p. 


CAMS 

ELEMENTARY  AND  ADVANCED 


BY 

FRANKLIN  DERONDE  FURMAN,  M.E. 

Professor  of  Mechanism  and  Machine  Design 

at  Stevens  Institute  of  Technology 
Member  of  American  Society  of  Mechanical  Engineers 


ELEMENTARY  CAMS 
FIRST  EDITION  (THIRD  IMPRESSION) 

TOTAL  ISSUE  FOUR  THOUSAND 

CAMS— ELEMENTARY  AND  ADVANCED 
FIRST  EDITION 


NEW   YORK 

JOHN  WILEY  &  SONS,  Inc. 

LONDON:  CHAPMAN  &  HALL,  LIMITED 
1921 


Copyright,  1916,  1921 

BY 
FRANKLIN  DzRONDE  FURMAN 


,,J, 


or 

BRAUNWORTH    &    CO. 

BOOK    MANVTACTURERi 

BROOKLYN.     N.     Y. 


Engineering 
Library 


PREFACE  TO  THE  ENLARGED  EDITION 

THE  first  five  sections  of  this  book  were  published  about  three 
years  ago  under  the  title  of  "Elementary  Cams."  The  chief  features 
of  this  earlier  book  were  that  it  pointed  out  a  classification,  an  arrange- 
ment, and  a  general  method  of  solution  of  the  well-known  cams  in 
such  manner  as  has  been  generally  developed  in  other  specialized 
branches  in  technical  engineering  work;  and  also  it  gave  a  series  of 
cam  factors  for  base  curves  in  common  use,  which  enabled  designers 
to  compute  proper  cam  sizes  for  specific  running  conditions,  offering 
numerous  examples  in  the  use  of  these  factors  in  the  several  kinds  of 
cam  problems.  The  factors,  with  the  exception  of  the  one  for  the 
30°  pressure  angle  for  the  Crank  Curve  were  new,  so  far  as  the 
author  is  aware.  The  "  Elementary  Cams "  will  continue  to  be 
sold  as  a  separate  volume. 

A  further  development  of  the  subject  is  given  in  the  present  work 
which  is  under  the  title  of  "Cams."  The  chief  original  features  of 
this  advanced  work  include  the  development  or  use,  or  both,  of  the 
logarithmic,  cube,  circular,  tangential  and  involute  base  curves, 
the  establishing  of  cam  factors  for  such  of  these  curves  as  have 
general  factors,  and  the  demonstration  that  the  logarithmic  base 
curve  gives  the  smallest  possible  cam  for  given  data. 

The  new  material  now  introduced  into  the  book  includes,  further, 
comparisons  of  the  characteristic  results  obtained  from  all  base 
curves,  in  which  the  relative  size  of  each  cam,  and  the  relative  velocity 
and  acceleration  produced  by  each,  is  shown  graphically  in  one 
combined  group  of  illustrations,  thus  enabling  the  designer  to  glance 
over  the  entire  field  of  theoretical  cam  design  and  quickly  select  the 
type  that  is  best  adapted  for  the  work  in  hand.  From  these  diagrams 
one  may  observe,  for  example,  which  form  of  cam  is  best  adapted  for 
gravity,  spring  or  positive  return,  which  is  best  for  slow  or  fast  veloci- 
ties at  various  points  in  the  stroke,  and  which  ones  are  apt  to  develop 
"hard  spots"  in  running.  The  involute  curve  is  found  to  have  its 
chief  and  characteristic  theoretical  advantage  when  it  is  used  with 
an  offset  follower.  The  nature  of  the  contact  between  cylindrical, 
conical  and  hyperboloidal  roller  pins,  when  used  in  connection  with 
grooved  cylindrical  cams,  has  been  investigated  and  pointed  out. 
The  subject  of  pure  rolling  contact  between  various  forms  of  oscillat- 
ing cam  arm  surfaces,  and  of  the  nature  and  amount  of  sliding  action 

iii 


iv  PREFACE    TO    THE    ENLARGED    EDITION 

of  such  surfaces  has  been  developed  so  that  the  effects  of  wear  due 
to  rubbing  may  be  confidently  considered  when  such  types  of  cams 
are  under  design. 

While  the  whole  purpose  of  this  work  has  been  to  present  the 
subject  matter  in  graphical  form  and  in  the  simplest  possible  manner 
so  as  to  make  it  available  to  the  greatest  number,  much  mathematical 
investigation  has  been  necessary  and  in  this  I  have  been  greatly 
aided  by  my  colleague  Professor  L.  A.  Hazeltine,  M.  E.,  head  of  the 
department  of  Electrical  Engineering  at  Stevens,  to  whom  I  express 
my  deep  appreciation.  The  details  of  these  investigations  are  not 
necessary  here  and  are  not  set  down,  but  their  results  are.  These 
results  are  given  in  various  formulas  that  are  used  in  the  solution  of  a 
number  of  the  problems.  These  final  formulas  avoid  the  use  of 
calculus  and  are  mostly  in  such  form  as  to  be  readily  used  by  designers 
generally. 

In  closing,  the  author  desires  to  introduce  a  personal  thought 
that  has  grown  up,  and  which  is  inseparable,  with  this  book.  Some 
years  ago,  before  any  special  study  was  given  by  the  writer  to  the  sub- 
ject of  cams,  it  appeared  that  the  whole  subject  of  mechanism  was  so 
thoroughly  covered  by  various  text  books  and  technical  papers  that 
the  time  in  engineering  development  had  arrived  when  there  was 
but  little  for  an  instructor  to  look  forward  to  in  the  way  of  produc- 
tion of  extended  original  work  on  any  given  topic.  To  say  the  least 
such  a  thought  was  not  at  all  encouraging,  and  so  it  is  a  pleasure  now 
to  the  author,  and  it  is  hoped  that  it  will  be  an  inspiration  particularly. 
to  the  younger  readers,  to  record  that  the  study  of  this  subject  of 
cams  has  brought  forth  a  great  wealth  of  new  and  practical  material 
which  had  not  previously  been  brought  to  light  and  set  down  in  the 
literature  of  the  subject.  Now  that  this  work  is  done,  the  vastness 
of  the  "  unknown,"  even  in  this  present  era  of  great  accomplish- 
ments, is  realized  as  it  never  was  before,  and  it  only  remains  to  suggest 
that  not  only  this  topic  of  cams  but  many  other  topics  in  the  science 
of  engineering  may  offer  opportunities  for  much  further  development 
and  perfection  on  the  part  of  those  who  have  the  desire  for  such  work 
and  the  time  to  pursue  it. 

F.  DER.  FURMAN. 
HOBOKEN,  N.  J.,  April,  1920. 


CONTENTS 

PAGES 

SECTION  I. — DEFINITIONS  AND  CLASSIFICATION 1-19 

Cams  Follower  Surfaces  Radial  or  Disk  Cams  Side  or 
Cylindrical  Cams  Conical  and  Spherical  Cams 

Names  of  Cams — Periphery,  Plate,  Heart,  Frog,  Mushroom,  Face, 
Wiper,  Rolling,  Yoke,  Cylindrical,  End,  Double  End,  Box,  Internal, 
Offset,  Positive  Drive,  Single  Acting,  Double  Acting,  Step,  Adjustable, 
Clamp,  Strap,  Dog,  Carrier,  Double  Mounted,  Multiple  Mounted, 
Oscillating 

Definitions  of  Terms  Used  in  the  Solution  of  Cam  Problems — Cam 
Chart,  Cam  Chart  Diagram,  Time  Chart,  Base  Curve,  Base  Line, 
Pitch  Line,  Pitch  Circle,  Pitch  Surface,  Working  Surface,  Pitch  Point, 
Pressure  Angle 

Formula  for  Size  of  Cam  for  a  Given  Maximum  Pressure  Angle 
Table  of  Cam  Factors  for  All  Base  Curves  for  Maximum  Pressure 
Angles  from  20°  to  60° 

SECTION  II. — METHOD  OF  CONSTRUCTION  OF  BASE  CURVES  IN  COMMON 

USE 20-24 

Straight  Line  Base  Straight-Line  Combination  Curve  Crank 
Curve  Parabola  Elliptical  Curve 

SECTION  III. — CAM  PROBLEMS  AND  EXERCISE  PROBLEMS       25-74 

Problem  1,  Empirical  Design  Problem  2,  Technical  Design. 
Advantages  of  Technical  Design  Problem  3,  Single-Step  Radial 
Cam,  Pressure  Angle  Equal  on  Both  Strokes  Omission  of  Cam 
Chart  Problem  4,  Single-Step  Radial  Cam,  Pressure  Angles  Unequal 
on  Both  Strokes 

Pressure  Angle  Increases  as  Pitch  Size  of  Cam  Decreases  Change 
of  Pressure  Angles  in  Passing  from  Cam  Chart  to  Cam  Cam  Con- 
sidered as  Bent  Chart.  Base  Line  Angles  Before  and  After  Bending 

Limiting  Size  of  Follower  Roller  Radius  of  Curvature  of  Non- 
Circular  Arcs 

Problem  5,  Double-Step  Radial  Cam  Determination  of  Maximum 
Pressure  Angle  for  a  Multiple-Step  Cam 

Problem  6,  Cam  with  Offset  Roller  Follower  Problem  7,  Cam 
with  Flat  Surface  Follower  Limited  Use  of  Cams  with  Flat  Surface 
Followers 

Problem  8,  Cam  with  Swinging  Follower  Arm,  Roller  Contact — 
Extremities  of  Swinging  Arc  on  Radial  Line  Problem  9,  Cam 
with  Swinging  Follower  Arm,  Roller  Contact — Swinging  Arc,  Con- 
tinued, Passes  Through  Center  of  Cam  Effect  of  Location  of 
Swinging  Follower  Arm  Relatively  to  the  Cam 


VI  CONTENTS 

PAGES 

Problem  10,  Face  Cam  with  Swinging  Follower  Problem  11, 
Cam  with  Swinging  Follower  Arm,  Sliding  Surface  Contact  Data 
Limited  for  Followers  with  Sliding  Surface  Contact 

Problem  12,  Toe  and  Wiper  Cam  Modifications  of  the  Toe 
and  Wiper  Cam 

Problem  13,  Single  Disk  Yoke  Cam  Limited  Application  of 
Single  Disk  Yoke  Cam  Problem  14,  Double  Disk  Yoke  Cam 

Problem  15,  Cylindrical  Cam  with  Follower  that  Moves  in  a 
Straight  Line  Refinements  in  Cylindrical  Cam  Design  Prob- 
lem 16,  Cylindrical  Cam  with  Swinging  Follower  Chart  Method 
for  Laying  Out  a  Cylindrical  Cam  with  a  Swinging  Follower  Arm 

Exercise  Problems,  3a  to  16a 

SECTION  IV. — TIMING  AND  INTERFERENCE  OF  CAMS 75-  78 

Problem  17,  Cam  Timing  and  Interference  Location  of  Key- 
ways  Exercise  Problem  17a 

SECTION  V. — CAMS  FOR  REPRODUCING  GIVEN  CURVES  OR  FIGURES  .      .     79-  87 

Problem  18,  Cam  Mechanism  for  Drawing  an  Ellipse  Prob- 
lem 18a,  Exercise  Problem  for  Drawing  Figure  8  Problem  19, 
Cam  for  Reproducing  Handwriting  Using  Script  Letters  Ste 

Method  of  Subdividing  Circles  into  Any  Desired  Number  of  Equal 
Parts 

SECTION  VI. — ADVANCED  GROUP  OF  BASE  CURVES 88-137 

Complete  List  and  Comparison  of  Base  Curves,  Their  Appli- 
cations and  Characteristic  Motions  Velocity  and  Acceleration 
Diagrams  Showing  Characteristic  Action  of  Various  Cams  All- 
logarithmic  Curve  Gives  Smallest  Possible  Cam  for  a  Given  Pres- 
sure Angle 

Problem  20,  All-logarithmic  Cam  General  Analysis  Detail 
Construction  of  Logarithmic  Curve  and  Cam  by  Analytical  and 
Graphical  Methods 

Problem  21,  Logarithmic-combination  Cam  with  Parabolic 
Easing-off  Arcs 

Problem  22,  Cam  with  Straight-Line  Base 

Straight-Line  Combination  Curve  Crank  Curve  Effect 
of  Crank  Curve  Following  Its  Tangent  Closely  Parabola 
Gravity  Curve  Curve  of  Squares  Perfect  Cam  Action 
Comparison  of  Parabola  and  Crank  Curves 

Problem  23,  Tangential  Cam,  Case  1  Graphical  and  Ana- 
lytical Methods  Characteristic  Retardation 

Problem  24,  Circular  Base  Curve  Cam,  Case  1  Elliptical  Base 
Curve  Effect  of  Varying  Axes  of  Ellipses  Elliptical  Base 
Curve  Equivalent  to  Nearly  All  Other  Base  Curves 

Cube  Curve  Symmetrically  and  Unsymmetrically  Applied 

Problem  25,  Cube  Curve  Cam,  Case  1 


CONTENTS 


PAGES 

Cams  Specially  Designed  for  Low-Starting  Velocities 
Problem  26,  Circular  Base  Curve  Cam,  Case  2 
Problem  27,  Cube  Curve  Cam,  Case  2 
Problem  28,  Tangential  Cam,  Case  2 

SECTION  VII. — CAM  CHARACTERISTICS 138-156 

Methods  of  Determining  Velocities  and  Accelerations  Time- 
Distance,  Time- Velocity  and  Time-Acceleration  Diagrams 
Degree  of  Precision  Obtained  by  Graphical  Methods  Comparison 
of  Relative  Velocities  and  Forces  Produced  by  Cams  of  Different 
Base  Curves  Cam  Follower  Returned  by  Springs  Rela- 
tive Strength  of  Spring  Required  for  Cams  of  Different  Base  Curves 
Special  Adaptation  of  Cube  Curve  Cam  for  Follower  Returned 
by  a  Spring  Nature  of  Pressure  between  Cam  Surface  and 
Spring-Returned  Follower 

Accuracy  in  Cam  Construction 

Regulation  of  Noise        High-Speed  Cams        Balancing  of  Cams 

Pressure  Angle  Factors,  Nature  of  Application  and  Method  of 
Determination  for  All  Base  Curves  Varied  Forms  of  Funda- 
mental Base  Curves  Chart  Showing  Values  of  Intermediate 
Pressure  Angles  from  20°  to  60°  for  all  Cams 

SECTION  VIII. — MISCELLANEOUS  CAM  ACTIONS  AND  CONSTRUCTIONS  .   157-229 

Variable  Angular  Velocity  in  Driving  Cam  Shaft 

Problem  29,  Oscillating  Cam  Having  Variable  Angular  Velocity, 
Toe  and  Wiper  Type 

Problem  30,  Wiper  Cam  Operating  Curved-Toe  Follower 

Problem  31,  Sliding  Action  between  Cam  and  Flat  Follower  Sur- 
faces Rate  of  Sliding  Measured  Velocity  of  the  Follower 
Measured 

Problem  32,  Sliding  Action  between  Cam  and  Curved-Toe  Fol- 
lower 

Problem  33,  Sliding  Action  where  Driving  Cam  Has  Variable 
Angular  Velocity 

Elimination  of  All  Sliding  Action  between  Cam  and  Flat  or 
Curved  Surface  Follower 

The  Princip  e  of  Pure  Rolling  Action  between  Cam  Surfaces 
Well-Known  Curves  that  Lend  Themselves  Readily  to  Pure 
Rolling  Cam  Action 

Problem  31,  Pure  Rolling  with  Flat-Surfacs  Follower 

Use  of  Logarithmic  Curve  for  Pure  Rolling  Action  Charac- 
teristic Properties  of  the  Logarithmic  Curve 

Problem  35,  Pure  Rolling  with  Logarithmic  Curved  Cam  Arm 
Angular  Motion  of  Each  Arm  Tangency  of  Logarithmic  Cam 
Surfaces  Regulation  of  Pressure  Angle  when  logarithmic 
Rolling  Cams  are  Used 

Derived,  or  Computed  Curves  for  Rolling  Cam  Arms 


Vlll  CONTENTS 

PAGES 

Problem  36,  The  Use  of  a  Derived  Curve  for  Rolling  Cam  Arms          174 

Rolling  Cam  Arms  Useful  for  Starting  Shafts  Gradually 

Regulation  of  Pressure  Angle  with  Derived  Rolling  Cams 

Elliptical  Arcs  for  Pure  Rolling  Cam  Arms 

Problem  37,  Elliptical  Rolling  Cam  Arms,  Angles  of  Action  Equal 
Determination  of  Major  and  Minor  Axes  of  Ellipses  Construc- 
tion of  Ellipse  Pressure  Angle  in  Rolling  Elliptical  Cam  Arms 

Problem  38,  Elliptical  Rolling  Cam  Arms,  Angles  of  Action 
Unequal 

Pure  Rolling  Parabolic  Cam  Surfaces  for  a  Reciprocating  Motion 

Problem  39,  Rolling  Parabolas  Construction  of  Parabola 
Pure  Rolling  Hyperbolic  Cam  Arms  where  Centers  are  Close 
Together 

Problem  40,  Rolling  Hyperbolas        Construction  of  Hyperbola 

Detail  Drawing  of  Cylindrical  Cams  The  True  Maximum 
Pressure  Angle  in  Cylindrical  Cams  Drawing  of  Groove  Out- 
lines, Approximate  and  More  Exact  Methods 

Forms  of  Follower  Pins  for  Cylindrical  Grooved  Cams  Line  of 
Contact  between  Pin  and  Groove  Surface,  at  Rest  and  Moving 
The  Cylindrical  Follower  Pin  The  Conical  Follower  Pin  The 
Hyperboloidal  Follower  Pin 

Plates  for  Cylindrical  Cams  Adjustable  Cylindrical  Cams  for 
Automatic  Work 

Double-Screw  Cylindrical  Cams  Periods  of  Rest  of  More 
than  One  Revolution  in  Cylindrical  Cams  Slow-advance  and 
Quick-Return  Secured  by  Double-Screw  Cam 

Straight-Sliding  Plate  Cams 

Involute  Cams  Construction  of  Involute  Curve  Pressure 
Angle  with  Involute  Cam 

Involute   Cam    Specially    Adapted    for    Flat-Surface    Follower 

Problem  41,  Involute  Cam  with  Radial  Follower 

Oscillating  Positive-Drive-Single-Disk  Cam  Cam  Shaft  Acting 
as  Guide  Positive  Drive  with  Cam  Shaft  as  Guide  Positive- 
Drive  Double-Disk  Radial  Cam  with  Swinging  Follower  Rotary- 
Sliding  Yoke  Cams  Giving  Intermittent  Harmonic  Motion,  and 
Reciprocating  Motion  Rotary  Sliding  Yoke  Cam,  General  Case 
Cam  Surface  on  Reciprocating  Follower  Rod 

Problem  42,  Definite  Motion  where  Cam  Surface  is  on  Follower 
Rod 

Problem  43,  Cam  Surface  on  Swinging  Follower  Arm 

Effect  of  Swinging  Transmitter  Arm  between  Ordinary  Radial 
Cam  and  Follower  Angular  Velocity  Curve  for  a  Swinging 
Follower  Arm  Velocity  Curve  for  a  Follower  Rod  with  Com- 
parison of  Results  Obtained  by  Using  Transmitter  Arms  with 
Sliding  and  Roller  Action  Diagram  of  Pressure  Angles  Meas- 
urement of  Rubbing  Velocities  in  Cams  Having  Sliding  Action 
Boundary  of  Follower  Surface  Subjected  to  Wear  in  Sliding  Cams  221 


CONTENTS  ix 

PAX5ES 

Cam  Action  Different  on  Forward  and  Return  Strokes  with  Sliding          222 
Cams 

Problem  44,  Small  Cams  with  Small  Pressure  Angles  Secured  by 
Using  Variable  Drive  Variable  Drive  by  Whitworth  Motion 

Swash  Plate  Cam  Uniformly  Rotating  Cam  Giving  Inter- 
mittent Rotary  Motion  The  Eccentric  a  Special  Type  of  Cam 
An  Example  of  a  Time-Chart  Diagram  for  Eleven  Cams  on  One 
Shaft  of  an  Automatic  Machine  229 


ELEMENTARY  CAMS 


SECTION   I.— DEFINITIONS   AND   CLASSIFICATION 

DEFINITIONS 

1.  CAMS  are  rotating  or  oscillating  pieces  of  mechanism  having 
specially  formed  surfaces  against  which  a  follower  slides  or  rolls 
and  thus  receives  a  reciprocating  or  intermittent  motion  such  as 
cannot  be  generally  obtained  by  gear  wheels  or  link  motions. 

Various  forms  of  cams  are  illustrated  at  C  in  Figs.  1  to  10.  The 
follower  in  each  case  is  shown  at  F,  all  having  roller  contact  except 
the  ones  shown  in  Figs.  7  and  8.  The  former  has  a  V  edge  and 
the  latter  a  plane  surface  in  contact  with  the  cam  and  both  have 
sliding  action. 

2.  FOLLOWER  EDGES  OK  ROLLERS  may  have  motion  in  a  straight 
line  as  from  D  to  G,  Fig.  7,  or  in  a  curved  path  depending  on  suit- 
ably constructed  guides  or  on  swinging  arms.     The  total  range  of 
travel   of  the  follower  may  be   accomplished   by  one   continuous 
motion,  or  by  several  separate  motions  with  intervals  of  rest.     Each 
motion  may  be  either  constant  or  variable  in  velocity,  and  the  time 
used  by  the  motion  may  be  greater  or  less,  all  according  to  the 
work  the  machine  has  to  do  and  to  the  will  of  the  designer. 

CLASSIFICATION 

3.  Cams  may  be  most  simply,  and  at  the  same  time  most  com- 
pletely, classified  according  to  the  motion  of  the  follower  with  re- 
spect to  the  axis  of  the  cam,  as: 

(a)  RADIAL  OR  DISK  CAMS,  in  which  the  radial  distance  from  the 
cam  axis  to  the  acting  surface  varies  constantly  during  part  or  all  of 
the  cam  cycle,  according  to  the  data.  The  follower  edge  or  roller 
moves  in  all  cases  in  a  radial,  or  an  approximately  radial,  direction 
with  respect  to  the  cam.  Various  forms  of  radial  cams  are  illus- 
trated in  Figs.  1,  2,  7,  8,  and  9. 

(6)  SIDE  OR  CYLINDRICAL  CAMS,  in  which  the  follower  edge  or 
roller  moves  parallel  to  the  axis  of  the  cam,  or  approximately  in 

1 


J^EMENTAHY   CAMS 


this  direction.     Several  types  of  side  cams  arc  shown  in  Figs.  3,  4, 
and  10. 

Nearly  all  the  cams  referred  to  in  the  above  figures  illustrating 
the  two  general  classes  of  radial  and  side  cams  respectively  have 
special  or  local  trade  names  which  will  be  pointed  out  in  a  succeed- 
ing paragraph. 

(c)  CONICAL  and  (d)  SPHERICAL  cams,  in  which  the  follower  edge 
or  roller  moves  in  an  inclined  direction  having  both  radial  and 

longitudinal  components 
with  respect  to  the  axis 
of  the  cam  as  illustrated 
in  Figs.  5  and  6. 

4.  NAMES  OF  CAMS. 
Cams,  in  popular  usage, 
have  come  to  be  known 
by  a  wide  range  of  names, 
the  same  cam  often  being 
designated  by  a  number 
of  different  names  accord- 
ing to  geographical  loca- 
tion and  personal  prefer- 
ence and  surroundings  of 
the  cam  builder  or  user. 
This  is  an  unfortunate  con- 
dition, and  in  the  general 
classification  in  the  preced- 
ing paragraph  an  endeavor 
is  made  to  establish  a  fun- 
damental basis  for  clarifying  and  simplifying  the  nomenclature 
of  cams  as  much  as  possible.  In  a  treatise  of  this  kind,  however,  it 
is  essential  that,  at  least,  the  more  common  of  the  ordinary  working 
terms  be  recognized  and  defined,  and  that  the  cams  under  their 
popular  names  be  properly  placed  in  the  fundamental  classification 
given  in  the  preceding  paragraph. 

The  following  specially  named  cams  fall  under  the  classifica- 
tion of  radial  cams: 

(e)  PERIPHERY  CAMS,  in  which  the  acting  surface  is  the  periphery 
of  the  cam,  as  illustrated  in  Figs.  1,  7,  and  9.  While  these  are  ex- 
amples of  true  periphery  cams,  it  must  be  recorded  that  the  cylin- 
drical grooved  cam,  shown  in  Fig.  3,  is  also  known  to  some  extent 
as  a  periphery  cam,  due  no  doubt  to  the  fact  that  in  designing  this 


END 


TRONT 


FIG.  1. — RADIAL  CAM  AND  FOLLOWER, 
ROLLER  CONTACT 


DEFINITIONS   AND    CLASSIFICATION  3 

cam  the  original  layout  for  the  contour  of  the  groove  is  first  made 
on  a  flat  piece  of  paper,  which  is  then  wrapped  on  to  the  surface  or 
"periphery"  of  the  cylinder.  Since  the  contour  line  of  the  groove 
which  lies  on  the  periphery  is  merely  a  guiding  line  for  cutting  the 
groove,  and  since  the  side  surface  of  the  groove  is  the  working  sur- 
face, it  is,  to  say  the  least,  a  misnomer  to  designate  such  a  cam  as 
a  periphery  cam. 

(/)  PLATE  CAMS,  in  which  the  working  surface  includes  the  full 
360°,  and  forms  either  the  periphery  of  the  cam,  or  the  sides  of  a 


FltONT 


FIG.  2. — FACE  CAM  AND  FOLLOWER 


groove  cut  into  the  face  of  the  cam  plate,  as  illustrated  in  Figs.  1 
and  2  respectively.  Figs.  7  and  9  also  show  plate  cams. 

(g)  HEART  CAMS,  in  which  the  general  form  is  that  which  the 
name  implies.  See  Fig.  7.  In  this  type  of  cam  there  are  two 
distinct  symmetrical  lobes,  often  so  laid  out  as  to  give  uniform 
velocity  to  the  driver.  In  this  case  each  lobe  would  be  bounded 
by  an  Archimedean  spiral  with  the  ends  eased  off. 

(h)  FROG  CAM,  in  which  the  general  form  includes  several  lobes 
more  or  less  irregular,  as  illustrated,  for  example  at  C  in  Fig.  9. 

(i)  MUSHROOM  CAM,  in  which  the  periphery  of  a  radial  or  disk 
cam  works  against  a  flat  surface,  usually  a  circular  disk  at  right 
angles  to  the  cam  disk,  instead  of  against  a  roller,  see  Fig.  44. 

(j)  FACE  CAM,  also  called  a  Groove,  but  more  properly  a  Plate 
Groove  cam,  to  distinguish  it  from  the  Cylindrical  Groove  cam,  in 
which  a  groove  is  cut  into  the  flat  face  of  the  cam  disk.  In 


4  ELEMENTARY   CAMS 

this  form  of  cam  shown  in  Fig.  2  the  roller  has  two  opposite  lines 
of  contact,  one  against  each  side  of  the  groove,  when  the  roller  has 
a  snug  fit.  The  plate  or  disk  in  which  the  groove  is  cut  is  generally 
circular;  but  it  may  be  cast  to  conform  with  the  contour  of  the 
groove,  or  it  may  be  built  with  radial  arms  supporting  the  irregular 
grooved  rim.  In  the  latter  case  it  lacks  resemblance  to  the  face 


END 

FRONT 
FIG.  3. — CYLINDRICAL  CAM  AND  SWINGING  FOLLOWER 

cam,  but  nevertheless  it  must,  because  of  the  nature  of  its  action, 
be  classed  with  it.  The  face  cam,  as  ordinarily  considered  and  as 
illustrated  in  Fig.  2,  is  better  adapted  for  higher  speeds  because  of 
its  more  nearly  balanced  form  of  construction.  Against  this,  how- 
ever, must  be  considered  one  of  two  disadvantages,  either  the  high 
rubbing  velocity  of  the  roller  against  one  side  of  the  groove  when 
the  roller  is  a  snug  fit,  or  lost  motion  and  noise  as  the  working  line 
of  contact  changes  from  one  side  of  the  groove  to  the  other  when 
the  roller  has  a  loose  fit.  The  most  important  advantage  of  the 
face  cam,  that  of  giving  positive  drive,  will  be  considered  in  para- 
graph 9.  The  term  groove  cam  might  be  applied,  with  advantage 
in  clearness  of  meaning,  to  such  face  cams  as  are  cut  or  cast  on 
non-circular  plates. 


DEFINITIONS   AND    CLASSIFICATION  5 

(k)  WIPER  CAM,  which  has  an  oscillating  motion,  and  is  con- 
structed usually  with  a  long  curved  arm  in  order  that  it  may  "wipe" 
or  rub  along  the  plane  surface  of  a  long  projecting  "toe,"  or  follower. 
The  wiper  cam  is  used  generally  to  give  motion  to  a  follower  which 
moves  straight  up  and  down  as  shown  from  F  to  Fr  in  Fig.  8.  This, 
however,  is  not  essential  and  the  follower  may  also  have  a  swinging 


TOP 


END 


FIG.  4. — END  CAM  AND  FOLLOWER 


motion.  The  disadvantage  of  sliding  friction,  which  is  inseparable 
from  the  wiper  cam,  is  balanced  to  some  extent  by  the  fact  that 
the  very  sliding  permits,  within  certain  range,  of  the  assignment  of 
specified  intermediate  velocities  between  the  starting  and  stopping 
points  which  cannot  be  obtained  with  similar  forms  of  cams  which 
have  pure  rolling  action. 

(I)  ROLLING  CAM,  which  greatly  resembles  the  wiper  cam  in 
general  appearance,  but  which  is  totally  different  in  principle,  for 
the  curves  of  the  cam  and  follower  surfaces  are  specially  formed  so 
as  to  give  pure  rolling  action  between  them.  The  rolling  cam  is 
specially  well  adapted  to  cases  where  both  driver  and  follower  have 
an  oscillating  motion  and  where  the  velocities  between  the  starting 
and  stopping  points  are  not  important  and  are  not  specified. 


6 


ELEMENTARY   CAMS 


(m)  YOKE  CAM,  a  form  of  radial  cam  in  which  all  diametral  lines 
drawn  across  the  face  and  through  the  center  of  rotation  of  the 
cam  are  equal  in  length.  This  form  of  cam  permits  the  use  of 
two  opposite  follower  rollers  whose  centers  remain  a  fixed  distance 
apart,  to  roll  simultaneously  on  opposite  sides  of  the  cam,  and  thus 
give  positive  motion  to  the  follower.  For  illustration,  see  Fig.  9. 


FIG.  5. — CONICAL  CAM  AND  RECIPRO- 
CATING FOLLOWER 


FIG.  6. — SPHERICAL  CAM  AND  SWINGING 
FOLLOWER 


Yoke  cams  may  be,  and  frequently  are,  made  of  two  disks  fixed 
side  by  side,  the  peripheries  being  complementary  to  each  other 
and  the  two  rollers  of  the  yoke  rolling  on  their  respective  cam  surfaces, 
as  shown  in  Fig.  56.  The  advantage  of  yoke  cams  is  that  they 
give  positive  motion  with  pure  rolling  of  the  follower  roller,  there 
being  contact  on  only  one  side  of  the  roller  in  contradistinction  to 
the  double  contact  of  thf  roller  which  exists  in  face  and  groove 
cams. 

5.  The  following  specially  named  cams  fall  under  the  general 
classification  of  side  cams. 

These  include  cams  that  have  been  made  from  blank  cylindrical 
bodies  by  using  a  rotary  end  cutter  with  its  axis  at  right  angles 
to  the  axis  of  the  cylinder  and  by  moving  the  axis  of  the  rotary 
cutter  parallel  to  the  axis  of  the  cylinder  while  the  cylinder  rotates. 
A  groove  of  desired  depth  is  thus  left  in  the  cylinder,  Fig.  3,  or  the 
end  of  a  cylindrical  shell  is  thus  milled  to  a  desired  form,  Fig.  4. 
A  side  cam  may  also  be  formed  by  screwing  a  number  of  formed 


DEFINITIONS  AND    CLASSIFICATION 


\ 


clamps  on  to  a  blank  cyl- 
inder, the  sides  of  the 
clamps  thus  acting  as  the 
working  surface  as  illus- 
trated in  Fig.  11.  All 
types  of  side  cams  may 
properly  be  considered 
as  derived  from  blank 
cylindrical  forms,  and, 
therefore,  the  name  ' 'cyl- 
indrical cam"  could  be 
regarded  as  synonymous 
with  side  cam;  but  gen- 
eral custom  has  limited 
the  use  of  the  term  cyl- 
indrical cam  to  the  "bar- 
rel" or  "drum"  type 
mentioned  below: 

(n)  CYLINDRICAL  CAM, 
also  called  Barrel  cam, 
Drum  cam,  or  Cylindrical 

Groove  cam,  in  which  the    FIG.  7. — HEART  CAM  AND  FOLLOWER,  SLIDING  CONTACT 

groove,  cut  around   the 

cylinder,  affords  bearing  surface  to  the  two  opposite  sides  of  the 

follower  roller,  thus  giving  positive  motion,  as  illustrated  in  Fig.  3. 

(o)  END  CAM,  in  which  the  working  surface  has  been  cut  at  the 

end  of  a  cylindrical  shell,  thus  re- 
quiring outside  effort  such  as  a 
spring  or  weight  to  hold  the  follower 
roller  against  the  cam  surface  during 
the  return  of  the  follower.  An  end 
cam  is  shown  in  Fig.  4. 

(p)  DOUBLE  END  cam,  in  which 
a  projecting  twisted  thread  has  been 
left  on  a  cylindrical  body,  against 
both  sides  of  which  separate  rollers 
on  a  follower  arm  may  operate, 
and  thus  secure  positive  motion. 
Instead  of  cutting  down  a  cylinder 
to  leave  a  projecting  twisted  thread, 
FIG.  8.— TOE  AND  WIPER  CAM  it  may  be  cast  integral  with  a 


8 


ELEMENTARY   CAMS 


warped  plate,  as  illustrated  in  Fig.  10,  but  this  in  no  way  changes 
its  characteristic  action. 

There  are  a  number  of  names  in  common  use  for  cams,  that 
cover  both  radial  and  side  cams.  Most  prominent  in  this  connection 
are  those  mentioned  in  paragraphs  6  to  14. 

6.  Box  CAM,  which  designates  a'  cam  in  which  the  follower  roller 
is  encased  between  two  walls  as  in  the  face  cam,  Fig.  2,  or  the  cylin- 
drical cam,  Fig,  3.     Literally,  box  cams  would  also  include  yoke 
cams,  in  which  the  yoke  would  be  the  "box."     Box  cams,  because  of 
their  form  of  construction,  give  a  positive  drive  in  all  cases. 

7.  INTERNAL  CAM,  in  which  there  is  only  one  working  surface, 
and  this  is  outside  of  the  pitch  surface.     The  internal  cam  cor- 
responds to  the  internal  gear  wheel  in  toothed  gearing.     It  may  also 
be  considered  as  a  face  cam  with  the  inside  surface  of  the  groove 
removed,  thus  requiring  that  the  follower  roller  should  always  be  in 
pressure  contact  on  the  outside  surface  of  the  groove  by  means  of  a 
spring  or  weight,  etc.     Under  some  conditions  of  structural  arrange- 
ments of  the  cam  machine,  the  internal  cam  may  be  used  to  advan- 
tage where  it  will  give  a  positive  motion  to  a  follower  on  the  opposite 
stroke  to  that  of  the  periphery  cam;    and  it  will  also  sometimes 


FIG.  9. — YOKE  CAM 


permit  of  a  larger  roller  than  the  periphery  cam,  as  explained  in 
paragraphs  56  and  62. 

8.  OFFSET  CAM,  in   which  the  line  of  action  of  the  follower, 
when  extended,    does   not   pass   through  the   center   of   the   cam, 
see  Fig.  43. 

9.  POSITIVE-DRIVE  CAM  is  one  in  which  the  cam  itself  drives 
the  follower  on  the  return  as  well  as  the  forward  motion.     Most 


DEFINITIONS   AND    CLASSIFICATION 


9 


cams  drive  only  on  the  forward  motion  of  the  follower  and  depend 
upon  gravity  or  the  action  of  a  spring  to  drive  the  follower  in  its 
return  motion;  such  cams  are  illustrated  in  Figs.  1,  4,  5,  6,  7,  and  8. 
Cams  having  positive  drive,  and  therefore  independent  of  gravity 
or  springs,  are  illustrated  in  Figs.  2,  3,  9,  and  10.  It  will  be  noted 
that  positive-drive  cams  include  the  face,  yoke,  cylindrical,  and 
double-end  types  of  cams;  also  that  the  box  cam,  although  it  in- 
cludes some  of  these,  should  also  be  considered  as  a  group  name  of 
the  positive-drive  type. 

10.  SINGLE-ACTING  AND  DOUBLE-ACTING  CAMS  comprise  all  forms 
of  cams,  the  single-acting  ones  giving  motion  only  in  one  direction 
and  depending  on  a  spring  or  gravity  to  return  the  follower.     Double- 
acting  cams  have  the  follower  under  direct  control  all  the  time  and 
are  the  same  as  positive-drive  cams  described  in  the  preceding 
paragraph. 

11.  STEP  CAMS.     Cams  which  give  continuous  motion  to  the 


FRONT 


FIG.  10. — DOUBLE-END  CAM 

follower  from  one  end  of  the  stroke  to  the  other  are  called  single- 
step  cams.  When  the  follower's  motion  in  either  of  its  two  general 
directions  is  made  up  of  two  entirely  separate  movements  it  is  called 
a  double-step  cam  with  reference  to  that  stroke.  If  three  or  more 
separate  movements  are  given  to  the  follower  while  it  moves  in  one 
general  direction  it  is  generally  referred  to  as  a  multiple  step  cam, 
or  as  a  triple-step,  quadruple-step  cam,  etc.  Since  a  cam  may  be, 
for  example,  a  double-step  cam  on  the  out  or  working  stroke,  and 


EBONT; 


FIG.  11. — BAKREL  CAM 


FRON,T  END 

FIG.  12. — ADJUSTABLE  PLATE  CAM 


FRONT  END 

FIG.  13. — DOG  CAM 


DEFINITIONS   AND   CLASSIFICATION  11 

a  single-step  cam  on  the  return  stroke,  such  a  cam  may  be  referred 
to  as  a  two-one  step  cam,  always  giving  the  number  referring  to  the 
working  stroke  first. 

12.  ADJUSTABLE  CAM,  ALSO  CALLED  CLAMP  CAM,  STRAP  CAM,  DOG 
CAM,  AND  CARRIER  CAM,  in  which  specially  formed  pieces  are  directly 
bolted  or  clamped  to  any  of  the  regular  geometrical  surfaces,  usually 
to  either  the  plane  or  cylindrical  surfaces.     In  Fig.  12  the  clamps 
are  shown  at  C  and  D  fastened  to  a  disk.     The  cam,  considered  as  a 
whole,  belongs  to  the  radial  class.     In  Fig.  13  the  clamps  are  shown 
at  C  and  D,  also  fastened  to  a  disk,  but  in  this  case  the  clamps,  or 
dogs,  as  they  are  usually  called  when  used  in  this  way,  are  so  formed 
as  to  give  a  sidewise  motion  to  the  follower,  and  therefore  this  cam 
belongs  to  the  side  cam  class.     In  Fig.  11  clamps  are  shown  at  C, 
D,  E,  and  F  fastened  to  a  cylinder,  and  they  are  shaped  to  give  the 
same  action  as  a  regularly  formed  end-cam  in  the  side-cam  class. 
The  type  of  cam  illustrated  in  Fig.  11  is  also  known  as  an  adjustable 
cylindrical  or  " barrel"  or  "drum"  cam  and  is  very  widely  used  for 
regulating  the  feeding  of  the  stock,  and  in  operating  the  turret  in 
automatic  machines  for  the  manufacture  of  screws,  bolts,  ferrules, 
and  small  pieces  generally  that  are  made  up  in  quantities. 

13.  DOUBLE-MOUNTED  OR  MULTIPLE-MOUNTED  CAMS   are   some- 
times resorted  to  where  several  movements  can  be  concentrated 
into  small  space.     This  consists  merely  in  placing  two  or  more  of 
any  of  the  cam  surfaces  described  in  the  preceding  paragraphs  on 
one  solid  casting  or  cam  body.     For  example,  a  face  cam,  a  cylin- 
drical, and  an  end  cam  may  all  be  cut  on  one  piece. 

14.  OSCILLATING  CAMS,  in  which  the  cam  itself  turns  through  a 
fraction  of  a  turn  instead  of  through  the  entire  360°.     While  any 
type  of  cam  may  be  designed  to  oscillate  instead  of  rotate,  it  is 
usually  the  toe-and-wiper  and  rolling  forms  of  the  radial  type  of 
cam  that  are  known  as  oscillating  cams.     With  oscillating  cams  the 
follower  may  move  forth  and  back  on  a  straight  line,  or  it  may 
oscillate  also. 

15.  Cams  falling  in  the  conical  class  have  no  special  name  other 
than  the  one  here  used.     The  spherical  cams  are  sometimes  termed 
globe  cams.     Cams  in  conical  and  spherical  classes  are  particularly 
useful  in  changing  direction  of  motion  in  close  quarters  and  in 
directions  other  than  at  right  angles.     In  both  Figs.  5  and  6,  end 
action  of  the  cam  is  shown,  but  it  is  apparent  that  with  thicker  walls 
on  both  the  cone  and  the  sphere,  grooves  could  be  cut  in  them, 
thus  giving  positive  driving  cams  in  both  cases. 


12 


ELEMENTARY   CAMS 


16.  Summing  up  the  general  and  special  names  for  cams  we 
have  in  tabular  form: 


Cams 


c.  Periphery 

/  Plate 

g  Heart 

Box 
Internal 

a  Radial 
or  Disk 

h  Frog 
i   Mushroom 
j  Face  or  Plate  Grooved 

Offset 
Positive  Drive 
Single  Acting 
Double  Acting 

Oi 

k  Toe  and  Wiper 
I   Rolling 
m  Yoke  or  Duplex 
n  Cylindrical,  Grooved,  Barrel,  or 

Step 
Adjustable  or 
Strap 
Dog  or  Carrier 

b  Side,  or 
Cylindrical 

o  End 
p  Double  End 

Multiple 

Mounted 

Oscillating 

/»     (^rvnir»a1 

d  Spherical 
or  Globe 

DEFINITIONS  OF  TERMS  USED  IN  THE  SOLUTION  OF  CAM  PROBLEMS 

17.  CAM  CHART.  Illustrated  in  Fig.  14.  The  chart  is  a  rectangle 
the  height  of  which  is  equal  to  the  total  motion  of  the  follower  in 
one  direction,  and  the  length  equal  to  the  circumference  of  the  pitch 
circle  of  the  cam.  The  chart  length  represents  360°  and  is  sub- 


!|       2 

x"" 

^ 

i^: 

Pitch  Line 

iiEESf 

^-Ss* 

««0°  l<r2<r30J4<r50J60J70°80t'900                                        180°                                       270°                                   3W 

FIG.  14. — CAM  CHART 

divided  into  equal  parts  marking  the  5°,  10°  .  .  .  points,  or  the  J4 
J4  .  .  .  points,  or  any  other  convenient  subdivision,  according  to 
the  requirements  of  the  problem.  On  the  cam  chart  are  drawn  the 
base  curve  and  the  pitch  line.  The  former  becomes  the  pitch  surface 
of  the  cam  and  the  latter  the  pitch  circle. 

18.  CAM  CHART  DIAGRAM.  Illustrated  in  Fig.  15.  The  cam 
chart  diagram  is  a  rectangle,  the  height  of  which  represents  the 
total  motion  of  the  follower  in  one  direction.  The  length  of  the 
diagram  represents  the  circumference  of  the  pitch  circle  of  the  cam. 


DEFINITIONS   AND    CLASSIFICATION 


13 


In  the  cam  chart  diagram  the  scales  for  drawing  the  height  and  the 
length  of  the  rectangle  are  totally  independent  of  each  other  and 
independent  also  of  the  scale  of  the  cam  drawing.  In  drawing  the 
diagram  no  scale  need  be  used  at  all,  and  the  entire  chart  diagram 
with  its  base  curve  and  pitch  line  may  be  drawn  entirely  freehand 
with  suitable  subdivisions  marked  off  entirely  "by  eye"  according 
to  the  requirements  of  the  problem.  The  base  curve  may  be  drawn 
roughly  as  a  curve  or  it  may  be  made  up  of  a  series  of  straight  lines. 
The  cam  chart  diagram  frequently  serves  all  the  purposes  of  the 
cam  chart.  It  saves  time,  and  permits  of  chart  drawings  being 


^ 

\^r 

X. 

Pitch  Line, 

^ 

X 

0                                 90°                                /fO°                               270°                         3601 

-*  Represents  kniff/i  of  circumference,  of  pifch  circle  ofcam^ 

FIG.  15. — CAM  CHART  DIAGRAM 

made  on  small  available  sheets  of  paper,  whereas  the  more  precise 
cam  chart  often  requires  large  sheets  of  paper  which  are  usually 
impracticable  and  unnecessary  in  many  circumstances. 

19.  TIME  CHARTS.  Illustrated  in  Figs.  16  and  17.  Time  charts 
are  the  same  as  cam  charts  or  cam  chart  diagrams,  and  are  con- 
structed in  the  same  way  as  described  in  the  two  preceding  para- 
graphs. The  term  "time  chart,"  however,  is  most  appropriately 
applied  to  problems  where  two  or  more  cams  are  used  in  the  same 
machine  and  where  their  functions  are  dependent  on  each  other. 


; 

? 

•y 

f 

s 

x 

\ 

\ 

-> 

s 

\ 
/ 

s 

" 

/ 

- 

' 

/ 

/ 

/ 

X 

\ 

£ 

^, 

\ 

\ 

\ 

c 

$ 

> 

9 

/ 

^ 

^ 

^ 

90°  180°  270°  360° 

FIG.  16. — TIME  CHART  DIAGRAM,  BASE  CURVES  SUPERPOSED 

The  time  chart  permits  of  allowances  being  made  for  avoiding 
possible  interference  of  the  several  moving  parts,  and  for  the  desired 
timing  of  relative  motions  for  each  part.  The  time  chart  contains 
two  or  more  base  curves  according  to  the  number  of  cams  used. 
When  the  base  curves  are  superposed  as  in  Fig.  16,  the  time  chart 
consists  of  a  single  rectangle  whose  height  is  equal  to  the  greatest 


14 


ELEMENTARY   CAMS 


follower  motion.  The  superposing  of  curves  and  lines  often  leads 
to  confusion  and  error,  and  it  is  better,  in  general,  that  the  time 
chart  should  consist  of  a  series  of  charts  or  rectangles  all  of  the 
same  length  and  one  directly  under  the  other  as  in  Fig.  17.  Where 
there  are  many  base  curves  it  is  desirable  to  separate  the  rectangles 


0°  90°  180°  270°  360° 

FIG.  17. — TIME  CHART  DIAGRAM,  BASE  CURVES  SEPARATED 

by  a  small  space  to  avoid  any  possibility  of  confusion  due  to  different 
base  curves  running  together.  In  many  cases  the  term  "time  chart 
diagram,"  or  "timing  diagram,"  will  be  more  appropriate  than  "time 
chart"  in  just  the  same  way  as  the  cam  chart  diagram  is  more  ap- 
propriate than  the  cam  chart. 

20.  BASE  CURVE.     Illustrated  in  Fig.  14.     A  base  curve  is  made 
up  of  a  series  of  smooth  continuous  curves,  or  a  combination  of 
curves  and  straight  lines,  which  represent  the  motion  of  the  follower, 
and  which  run  in  a  wave-like  form  across  the  entire  length  of  the 
cam  chart  or  diagram.     The  base  curve  of  the  cam  chart  becomes 
the  pitch  surface  of  the  cam. 

21.  BASE  LINE.     Illustrated  in  Fig.  15.    A  base  line  is  made  up 
of  a  series  of  inclined  straight  lines,  or  a  series  of  inclined  and  hori- 
zontal lines,  in  consecutive  order,  which  zigzag  across  the  entire 
length  of  the  chart.     The  base  line  when  used  on  the  cam  chart 
indicates  the  exact  motion  of  the  follower,  but  when  used  on  a  cam 
chart  diagram  it  is  merely  a  time-saving  substitute  for  the  drawing 
of  the  base  curve.     The  base  line  of  the  cam  chart  diagram  represents 
the  pitch  surface  of  the  cam. 

22.  NAMES  OF  BASE  CURVES  OR  BASE  LINES  IN  COMMON  USE, 
see  Figs.  18  and  19: 

1.  Straight  line  4.  Parabola. 

2.  Straight-line  combination       5.  Elliptical  curve. 

3.  Crank  curve. 


DEFINITIONS   AND    CLASSIFICATION 


15 


23.  PITCH  LINE.  Illustrated  in  Fig.  14.  A  pitch  line  is  a 
horizontal  line  drawn  on  the  cam  chart  or  diagram,  and  it  becomes 
the  pitch  circle  of  the  cam.  The  position,  or  elevation,  of  the  pitch 
line  on  the  chart  varies  according  to  the  base  curve  which  is  specified, 
and  according  to  the  data  of  the  problem.  For  cams  which  give  a 


1  Straight  Line 

2  Straight.Line  Combination 

3  Crank  Curve 

4  Parabola 

5  Elliptical  Curve 


FIG.  18. — COMPARISON  OF  BASE  CURVES  IN  COMMON  USE  SHOWING  VARYING  DEGREES 
OF  MAXIMUM  SLOPE  WHEN  DRAWN  IN  SAME  CHART  LENGTH 

continuous  motion  to  the  follower  during  its  entire  stroke,  or  throw, 
the  pitch  line  will  pass  through  the  point  on  the  base  curve  which 
has  the  greatest  slope,  starting  from  the  bottom  of  the  chart.  This 
does  not  apply  to  all  possible  base  curves,  but  it  does  apply  to  all 


-17-73- 


-2.27- 


1  Straight  Line 

2  Straight  Line  Combination 

3  Crank  Curve 

4  Parabola 

5  Elliptical  Curve 


FIG.  19. — COMPARISON  OF  BASE  CURVES  IN  COMMON  USE  SHOWING  UNIFORM  MAXIMUM 
SLOPE  OF  30°  WHEN  DRAWN  IN  CHARTS  OF  VARYING  LENGTH 

those  mentioned  in  the  preceding  paragraph,  a  minor  exception 
being  made  of  the  crank  curve  which  will  be  referred  to  in  para- 
graph 34.  When  the  cam  causes  the  follower  to  move  through  its 
total  stroke  in  two  or  more  separate  steps  the  position  of  the  pitch 
line  on  the  chart  must  be  specially  found  as  will  be  explained  in 
problem  5. 


16  ELEMENTARY   CAMS 

24.  PITCH  CIRCLE.     Illustrated  in  Fig.  20.     A  pitch  circle  is  drawn 
with  the  center  of  rotation  of  the  cam  as  a  center,  and  its  circumfer- 
ence is  equal  to  the  cam  chart  length.     Its  characteristic  is  that  it 
passes  through  that  point  A,  Fig.  20,  of  the  pitch  surface  of  the  cam 
where  the  cam  has  its  greatest  side  pressure  against  the  follower. 
This  applies  to  all  cams  in  which  the  center  of  the  follower  roller 
moves  in  a  straight  radial  line.     For  other  motions  of  the  follower 
roller,  and  for  flat-faced  followers,  the  pitch  circle  must  be  specially 
considered,  as  will  be  explained  in  some  of  the  problems  covering 
these  types. 

25.  PITCH  SURFACE.     Illustrated  in  Fig.  20.     The  pitch  surface 
of  a  cam  is  the  theoretical  boundary  of  the  cam  that  is  first  laid 
down  in  constructing  the  cam.     When  the  follower  has  a  V-shaped 
edge,  as  at  D  in  Fig.  7,  the  pitch  surface  coincides  with  the  working 
surface  of  the  cam.     When  the  follower  has  roller  contact,  as  in 
Fig.  20,  the  pitch  surface  passes  through  the  axis  of  the  roller  and 
the  working  or  actual  surface  of  the  cam  is  parallel  to  the  pitch 
surface  arid  a  distance  from  it  equal  to  the  radius  of  the  roller. 

26.  WORKING  SURFACE.     Illustrated  in  Fig.   20.     The  working 
surface  of  the  cam  is  the  surface  with  which  the  follower  is  in  actual 
contact.     It  limits  the  working  size  and  weight  of  cam.     For  exact 
compliance  with  a  given  set  of  cam  data,  the  cam  has  only  one 
theoretical  size  which  is  bounded  by  the  pitch  surface,  but  the 
working  size  may  be  anything  within  wide  limits  which  depend  on 
the  radius  of  the  follower  roller  and  the  necessary  diameter  of  the 
cam  shaft. 

The  working  surface  is  found  by  taking  a  compass  set  to  the 
radius  of  the  roller  and  striking  a  series  of  arcs  whose  centers  are 
on  the  pitch  surface.  Such  a  series  of  arcs  is  shown  in  Fig.  20  with 
their  centers  at  B,  A,  etc.  The  curve  which  is  an  envelope  to 
these  arcs  is  the  working  surface. 

27.  PITCH  POINT  OF  FOLLOWER.     Illustrated  in  Fig.  20.     The 
pitch  point  of  the  follower  is  that  point  fixed  on  the  follower  rod  or 
arm  which  is  always  in  theoretical  contact  with  the  pitch  surface 
of  the  cam.     If  the  follower  has  a  sharp  V-edge  the  pitch  point  is 
the  edge  itself.     If  the  follower  has  a  roller  end,  the  pitch  point 
is  the  axis  of  the  roller.     The  pitch  point  is  constantly  changing  its 
position  from  C  to  D  as  the  follower  moves  up  and  down. 

28.  PRESSURE   ANGLE.     Illustrated   in   Fig.    20.     The   pressure 
angle  is  the  angle  whose  vertex  is  at  the  pitch  point  of  the  follower 
in  its    successive    positions    and    whose    sides    are    the    direction 


DEFINITIONS   AND    CLASSIFICATION 


1? 


\ 


of    motion  of   the   pitch    point    and    the    normal    to    the    pitch 
surface. 

Pressure  angles  exist  when  the  surface  of  the  cam  presses  sidewise 
against  the  follower;  they  cause  bending  in  the  follower  arm  and 
side  pressure  in  the  follower  guide  and  in  the  bearings.  The  pres- 


FIG.  20. — SHOWING  NAMES  OF  SURFACES,  LINES,  AND  POINTS  OF  A  CAM 

sure  angle  is  constantly  varying  in  all  cams  as  the  follower  moves 
up  and  down,  except  where  a  logarithmic  spiral  is  used.  In  assign- 
ing cam  problems  the  maximum  permissible  pressure  angle  is  usually 
given.  In  Fig.  20  the  pressure  angle  is  zero  at  (7,  it  will  be  equal  to 
a  when  B  reaches  J,  and  will  be  a  maximum  when  A  reaches  K. 

29.  FORMULA  FOR  SIZE  OF  CAM  FOR  A  GIVEN  MAXIMUM  PRESSURE 
ANGLE.  The  radius  of  the  pitch  circle  of  the  cam  may  be  found 
directly  by  the  formula: 

360  J_ 

X    b     X</  X27r 

hf 
6        ..*.....      (1) 

1   .  1 


57.3 


or. 


=  .159 


hf 
e 


(2) 


18 


ELEMENTARY   CAMS 


in  which,  r  =  radius  of  pitch  circle  of  cam. 
h  =  distance  traveled  by  follower. 
/  =  factor  for  a  given  maximum  pressure  angle. 
b  =  angle,  in  degrees,  turned  by  cam  while  follower  moves 

distance  h. 
e  =  angle,  in  fraction  of  revolution,  turned  by  cam  while 

follower  moves  distance  h. 

30.  CAM  FACTORS  FOR  MAXIMUM  PRESSURE  ANGLE.  The  factors, 
or  value  of  /,  for  various  maximum  pressure  angles  for  cams  using 
the  several  base  curves  in  common  use  are: 

TABLE  OF  CAM  FACTORS 


Name  of  Base  Curve 

MAXIMUM  PRESSURE  ANGLE  AND  VALUES  OP  / 

20° 

30° 

40° 

50° 

60° 

Straight  line  

2.75 
3.10 
4.32 
5.50 
6.25 

1.73 

2.27 
2.72 
3.46 
3.95 

1.19 
1.92 

1.87 
2.38 
2.75 

.84 
1.77 
1.32 
1.68 
1.95 

.58 
1.73 
.91 
1.15 
1.35 

Straight-line  combination*  .  .  . 
Crank  curve 

Parabola  

Elliptical  curvef               .    . 

These  factors,  for  30°,  are  illustrated  in  Fig.  19  where  each  of 
'the  base  curves  is  given  such  a  length,  in  terms  of  the  height,  that 
they  will  all  have  the  same  maximum  slope.  The  values  given  in 
this  table  are  also  shown,  graphically,  in  Fig.  21,  thus  enabling  one 
to  find  the  proper  cam  factor  for  any  intermediate  pressure  angle 
between  20°  and  60°. 


*  For  case  where  easing  off  radius  equals  follower's  motion. 

t  For  case  where  ratio  of  horizontal  to  vertical  axes  of  ellipse  is  7  to  4. 


DEFINITIONS   AND    CLASSIFICATION 
B     L  DO     F 


19 


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CAM  FACTORS 

FIG.  21. — CHART  SHOWING  RELATION  BETWEEN  PRESSURE  ANGLES  AND  CAM  FACTORS 
FOR  THE  ORDINARY  BASE  CURVES 


SECTION   II.— METHOD   OF   CONSTRUCTION   OF  BASE 
CURVES   IN   COMMON   USE 


31.  DETAIL  CONSTRUCTION  OF  BASE   CURVES.     The  method  of 
constructing  the  several  base  curves  for  a  rise  of  one  unit  of  the 
follower  will  be  explained  in  the  succeeding  paragraphs.     The  curves 
will  be  constructed  to  give  a  pressure  angle  of  30°  by  selecting  factors 
from  the  30°  column  in  the  table  in  the  preceding  paragraph.     Should 
the  base  curve  for  any  other  pressure  angle  be  desired  the  factor 
should  be  taken  from  the  corresponding  column. 

32.  STRAIGHT-LINE  BASE.     Fig.  22.     Lay  off  A  B  equal  to  the 
follower  motion,  which  will  be  taken  as   1   unit  in  these  illustra- 
tions.    Multiply  this  by  the  factor  1.73  from  paragraph  30,  and 
lay  off  the  distance  A  R  equal  to  it.     Complete  the  parallelogram 
and  draw  the  diagonal.     This  will  be  the  straight  line  base  and  the 


-1.73- 


FIG.  22. — STRAIGHT  BASE  LINE 


FIG.  23. — STRAIGHT-LINE  COMBINATION  CURVE 


angle  R  A  C  will  be  30°.  A  R  will  be  the  pitch  line.  These  base 
lines  and  curves  are  laid  off  from  right  to  left  so  that  they  may  be 
used  in  a  natural  manner  later  on  in  laying  out  the  cam  so  that 
it  will  turn  in  a  right-handed  or  clockwise  direction. 

The  straight-line  base  gives  abrupt  starting  and  stopping  velocities 
at  the  beginning  and  end  of  the  stroke  and  causes  actual  shock  in 
the  follower  arm.  The  velocity  of  the  follower  during  the  stroke  is 
constant.  The  acceleration  at  starting  and  retardation  at  stopping  is 
infinite  and  is  zero  during  the  stroke. 

33.  STRAIGHT-LINE  COMBINATION  CURVE.  Fig.  23.  Construct 
the  rectangle  with  a  height  of  1  unit  and  a  length  of  2.27  units. 
With  B  and  R  as  centers  draw  the  arcs  A  E  and  C  N,  and  draw  a 
straight  line  E  N  tangent  to  them.  The  angle  FEN  will  then  equal 
30°  and  the  line  A  C  will  be  a  base  curve  made  up  of  arcs  and  a 

20 


CONSTRUCTION    OF   BASE    CURVES   IN    COMMON   USE 


21 


straight  line  combined  to  form  a  smooth  curve.  D  F  will  be  the 
pitch  line. 

The  straight-line  combination  curve,  being  rounded  off  at  the 
ends,  does  not  give  actual  shock  to  the  follower  at  starting  and  stop- 
ping, but  it  does  give  a  more  sudden  action  than  any  of  the  base 
curves  which  follow,  and  the  maximum  acceleration  and  retardation 
values  are  comparatively  larger. 

34.  CRANK  CURVE.  Fig.  24.  Construct  the  rectangle.  Draw 
the  semicircle  EG  C  and  divide  it  into  any  number  of  equal  parts. 
Six  parts  are  best  for  practice  work  for  this  curve,  but  in  general 
in  practical  work  the  greater  the  number  of  divisions  the  more 
accurate  will  be  the  curve  and  the  smoother  the  action  of  the  cam. 


N 2.72 >j 

FIG.  24. — CRANK  CURVE 

The  six  equal  divisions  of  the  semicircle  are  readily  obtained  by 
taking  G  as  a  center  and  F  C  as  a  radius  and  striking  arcs  at  1  and 
5,  then  with  R  and  C  as  centers  mark  the  points  2  and  4  respectively. 
Divide  the  length  of  the  chart  into  six  equal  parts,  as  at  H,  I,  E,  etc. 
From  these  points  drop  vertical  lines,  and  from  the  corresponding 
divisions  on  the  semicircle  draw  horizontal  lines,  giving  intersecting 
points,  as  at  K,  on  the  desired  crank  curve.  The  tangent  to  the 
curve  at  E  will  then  make  an  angle  of  30°  with  the  line  E  F.  The 
pitch  line  will  be  D  F. 

When  the  crank  curve  is  transferred  from  the  chart  to  the  cam 
it  gives  an  angl£  which  is  a  fraction  of  a  degree  greater  than  30° 
at  the  point  E  on  the  cam  in  practical  cases.  This  is  not  enough 
greater  to  warrant  the  special  computations  and  drawing  that  would 
be  necessary  to  be  exact.  Therefore  the  method  of  laying  out  the 
crank  curve  and  the  pitch  line,  as  given  above,  will  be  adhered  to 
in  this  elementary  consideration  of  cam  work,  because  of  its 
simplicity. 

The  crank  curve  gives  a  slightly  irregular  increasing  velocity 
to  the  follower  from  the  beginning  to  the  middle  of  its  stroke;  then 
a  decreasing  velocity  in  reverse  order  to  the  end  of  the  stroke.  The 


22 


ELEMENTARY    CAMS 


acceleration  diminishes  to  zero  at  the  middle  of  the  stroke  and  then 
increases  to  the  end.  The  maximum  acceleration  and  retardation 
values  are  much  less  than  for  the  straight-line  combination  curve, 
and  are  only  a  little  greater  than  for  the  parabola. 

35.  PARABOLA.  Fig.  25.  Construct  the  rectangle.  Draw  the 
straight  line.RS  in  any  direction  and  lay  off  on  it  sixteen  equal 
divisions  to  any  scale.  From  the  sixteenth  division  draw  a  line  to  F, 
the  middle  point  of  the  chart;  draw  other  lines  parallel  to  this 
through  the  points  9,  4,  and  1,  thus  dividing  the  distance  R  F  into 
four  unequal  parts  which  are  to  each  other,  in  order,  as  1 ,  3,  5,  and 
7.  From  these  division  points  draw  horizontal  lines,  and  from  H, 
I,  and  J  drop  vertical  lines.  The  intersecting  points,  as  at  K, 


FIG.  25. — PARABOLA 

will  be  on  the  desired  parabola.  The  points  H,  7,  and  J  divide  the 
distance  D  E  into  four  equal  parts. 

The  parabola  gives  a  uniformly  increasing  velocity  from  the 
beginning  to  the  middle  of  the  stroke;  then  a  uniform!}'  decreasing 
velocity  to  the  end.  The  acceleration  of  the  follower  is  constant 
during  the  first  half  of  the  stroke  and  the  retardation  is  constant 
during  the  last  half.  The  acceleration  and  retardation  values  are 
equal  and  are  less  than  the  maximum  value  of  any  of  the  other  base 
curves.  This  means  that  the  direct  effort  required  to  turn  a  positive- 
acting  parabola  cam  is  less  than  for  any  other  type  of  positive  cam. 

36.  To  better  understand  the  smooth  action  given  by  the  cam 
using  this  curve,  consider,  1st,  D  H  as  a  time  unit  during  which  the 
follower  rises  one  space  unit;  2d,  H  I  as  an  equal  time  unit  during 
which  the  follower  rises  three  space  units;  3d,  I  J  as  the  time  unit 
during  which  the  follower  rises  five  space  units,  etc.  Inasmuch  as 
the  follower  travels  two  units  further  in  each  succeeding  time  unit, 
it  gains  a  velocity  of  two  units  in  each  time  unit,  and  this  is  uniform 
acceleration. 

The  distance  from  F  to  C  would  be  divided  the  same  as  from 
F  to  R  and  points  on  the  part  of  the  curve  from  E  to  C  similarly 


CONSTRUCTION    OF    BASE    CURVES    IN    COMMON    USE 


23 


located.  This  curve  will  be  identical  with  E  A,  but  in  reverse  order, 
and  will  give  uniform  retardation.  The  tangent  to  the  curve  A  C 
at  the  point  E  will  make  an  angle  of  30°  with  E  F,  and  D  F  will 
be  the  pitch  line. 

Eight  construction  points  were  taken  in  developing  the  curve 
A  C.     Eight  points  will  be  sufficient  for  beginners  for  practice  work 


FIG.  26. — ELLIPTICAL  CURVE 

and  later  six  points  may  be  used.  When  using  six  points  only  nine 
equal  divisions  should  be  laid  out  on  the  line  R  S,  the  remaining 
construction  being  the  same  as  described  above,  except  that  D  E 
should  be  divided  into  three  parts  instead  of  four.  In  practical  work 
many  more  construction  points  should  be  used  for  accuracy  and 
smooth  cam  action. 

37.  ELLIPTICAL    CURVE.     Fig.    26.     Draw    rectangle    A  B  C  R. 


Draw  semi-ellipse  making  F  G  equal  to  -r  F  C. 


To  draw  the  ellipse, 


take  a  strip  of  paper  with  a  straight  edge  and  mark  fine  lines  at 
P,  T,  and  S,  Fig.  26a,  making  P  T  =  C  F  and  P  S  =  G  F.  Move 
the  strip  of  paper  so  that  S  will  always  be  on 
the  line  R  C,  and  T  on  the  line  F  G;  P  will  then 
describe  the  path  of  the  ellipse.  Having  the  semi- 
ellipse,  divide  the  part  R  G,  Fig.  26,  into  four 
equal  arcs  as  at  1,  2,  3.  This  is  quickest  done 
by  setting  the  small  dividers  to  a  small  space  of 
any  value  and  stepping  off  the  distance  from  R 
to  G.  Suppose  that  there  are  18.8  steps.  Set 
down  this  number  and  divide  it  into  four  parts, 
giving  4.7,  9.4,  and  14.1.  Then  again  step  off  the 
arc  from  R  to  G  witn  the  same  setting  of  the  dividers,  marking  the 
points  that  are  at  4.7,  9.4,  and  14.1  steps.  The  compass  setting 
being  small,  the  fractional  part  of  it  can  be  estimated  with  all  prac- 
tical precision.  Divide  D  E  into  four  equal  parts  as  at  H,  I,  J. 
Draw  vertical  lines  from  these  points  and  horizontal  lines  from  the 


FIG.  26a. — SHOWING 
METHOD  OF  DRAW- 
ING SEMI-ELLIPSE 


24  ELEMENTARY    CAMS 

corresponding  points  at  1,  2,  and  3.  The  intersections,  as  at  K, 
will  give  a  series  of  points  on  the  elliptical  base  curve.  The  curve 
E  C  is  similar  to  A  E  but  in  reverse  order.  The  tangent  to  the  curve 
at  E  makes  an  angle  of  30°  with  E  F,  and  D  F  is  the  pitch  line. 

The  elliptical  base  curve  gives  slower  starting  and  stopping 
velocities  to  the  follower  than  any  of  the  other  curves,  but  the  velocity 
is  higher  at  the  center  of  the  stroke.  The  acceleration  is  variable  and 
increases  to  the  middle  of  the  stroke,  where  its  maximum  value  is 
greater  than  that  of  the  crank  curve  but  less  than  that  of  the  straight- 
line  combination  curve.  The  retardation  values  decrease  in  reverse 
order  to  the  end  of  the  stroke. 


SECTION  III.— CAM   PROBLEMS    AND   EXERCISE 
PROBLEMS 

38.  PROBLEM    1.     EMPIRICAL   DESIGN.     Required  a  radial  cam 
that  will  operate  a  V-edge  follower: 

(a)  Up       3  units  while  the  cam  turns    90°. 

(b)  Down  2     "         "       "      "        "       60°. 

(c)  Dwell  "       "      "        "      120°. 

(d)  Down  1  unit       "       "      "        "       90°. 

39.  Applying  the  simplest  process  for  laying  out  cams,  it  is  only 
necessary,  in  starting,  to  assume  a  minimum  radius  C  D,  Fig.  27,  for 


FIG.  27. — EMPIRICAL  DESIGN  OF  CAM  FOR  DATA  IN  PROBLEM  1,  V-EDGE  FOLLOWER 

the  cam,  and  then  lay  off  the  given  or  total  distance  of  3  units  as 
at  D  B.  The  assigned  angle  of  90°  is  next  laid  off  as  at  D  C  A  and 
the  point  DI  marked  so  as  to  be  3  units  further  out  than  D.  Any 
desired  curve  is  then  drawn  through  the  points  D  and  DI  and  part 
of  the  cam  layout  is  completed.  The  same  operations  are  repeated 
for  obtaining  the  points  Z)2  and  D3  and  the  entire  cam  is  finished. 
If  the  follower  had  roller  contact  instead  of  V-edge  contact,  a 

25 


26 


ELEMENTARY   CAMS 


minimum  radius  C  D,  Fig.  28,  would  be  assumed  as  in  the  previous 
case,  and  D  would  be  taken  as  the  center  of  the  roller.  The  closed 
curve  D,  D4,  D\  .  .  .  would  be  obtained  as  before  and  another  closed 
curve  E,  EI  .  .  .  would  be  drawn  parallel  to  it  at  a  distance  equal 


FIG.  28 — EMPIRICAL  DESIGN  OF  CAM  FOR  DATA  IN  PROBLEM  1,  ROLLER  FOLLOWER 

to  the  assumed  radius  of  the  roller.  The  latter  closed  curve  would 
be  the  actual  outline  of  the  cam. 

The  closed  curve  E  E\  .  .  .  would  be  known  as  the  working  sur- 
face and  the  curve  D  Di  .  .  .  as  the  pitch  surface  of  the  cam.  In 
Fig.  27  the  pitch  and  working  surfaces  coincide  because  the  follower 
has  a  V-edge. 

40.  Cams  are  sometimes  designed  with  no  more  labor  than  that 
entailed  in  the  previous  preliminary  problem.  And  it  may  be  added 
that  where  one  has  had  a  sufficient  experience  good  practical  results 
may  be  obtained  by  following  only  this  simple  method. 

The  method  of  cam  construction  described  above,  however,  does 
not  enable  the  cam  builder  or  designer  to  hold  in  control  the  velocity 
or  acceleration  of  the  follower  rod  D  G  as  it  moves  up  its  3  units; 
nor  does  it  enable  him  to  know  the  variable  and  maximum  side  pres- 
sures which  exist  between  the  follower  rod  and  the  bearing  or  guide 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


27 


F,  Fig.  27,  as  the  rod  moves  up.  In  order  that  these  things  may 
be  known,  this  preliminary  problem  will  now  be  redrawn  with  ad- 
ditional specifications. 

41.  PROBLEM  2.    TECHNICAL  DESIGN.    Required  a  radial  cam  that 
will  operate  a  roller  follower: 

(a)  Up       3  units  while  the  cam  turns    90°. 

(b)  Down  2     "         "       "      "        "       60°. 

(c)  Dwell  "       "      "        "      120°. 

(d)  Down  1  unit       "       "      "        "       90°. 

(e)  The  follower,  in  all  its  motions,  shall  move  with  uniform 
acceleration  and  uniform  retardation. 

(f)  The  maximum  side  pressure  of  the  cam  against  the  follower 
rod  shall  be  40°. 

Items  (a),  (b),  (c),  and  (d)  are  the  same  as  in  Problem  1. 

42.  Inasmuch  as  this  problem  is  given  at  this  place  simply  to 
show  that  velocity  and  acceleration  and  side  pressure  can  always 
be  controlled  with  very  little  additional  labor  beyond  that  necessary 
for  the  simple  layout  shown  in  Fig.  28,  the  full  explanations  of  the 
formula  and  figures  used  will  not  be  given  here.     They  will  be  taken 
up  in  their  proper  order  in  subsequent  paragraphs.     For  this  problem 
the  only  necessary  computation  is: 


4.55  =  Radius  of  pitch  circle  = 


r       573^       570  3X2'38 
-  57.3  -y    -  57.3        9Q 

C  H,  Fig.  29. 

The  reference  letters,  h,  f, 
and  b  are  defined  in  paragraph 
29.  Lay  off  C  H  in  Fig.  29,  and 
then  lay  off  the-  follower  motion 
of  3  units  equally  distributed  on 
each  side  of  H,  as  at  H  B  and 
H  D.  Divide  D  H  into  nine 
equal  parts  and  take  the  first, 
fourth,  and  ninth  parts;  do  like- 
wise with  B  H.  Divide  the  90° 
angle  B  C  DI  into  six  equal 
parts  by  radial  lines  as  shown, 
and  swing  each  of  the  six  di- 
vision points  between  D  and  B 

around   until  they  meet  SUCCes-    FlG-  SO.-TECHNICAL  DESIGN  OF  CAM  FOR  DATA 
.    *         TIT  IN    PROBLEM    2,    DRAWN  TO  SAME  SCALE 

sively     the    six    radial    lines.      A9  Fla.  2g 


28 


ELEMENTARY   CAMS 


A  curve  through  the  intersecting  points  will  be  the  pitch  surface  of 
the  cam,  as  shown  by  the  dash-and-dot  curve  D  HI  D\.  .  .  . 

The  working  surface  will  be  E  EI 
.  .  .  which  is  found  as  described 
in  paragraph  26. 

The  pitch  surface  DI  D2  is 
obtained  in  the  same  way  as 
D  DI  was  found.  The  curve 
D2  Z>3  is  an  arc  of  a  circle,  and 
the  curve  D$  D  is  found  in  the 
same  manner  as  D  DI. 

43.  ADVANTAGES     OF     THE 

TECHNICAL    DESIGN.      With    the 

cam  constructed  as  above  the 
follower  will  start  to  move  with 
the  same  characteristic  motion 
as  has  a  falling  body  starting 

FIG.  29.-(Duplicate)  TECHNICAL  DESIGN  OF   from  rest    an(J  the  follower  will 
CAM  FOR  DATA  IN  PROBLEM  2,  DRAWN  TO  . 

SAME  SCALE  AS  FIG.  28  be  stopped  with  the  same  gen- 

tle motion  in  reverse  order.    It 

will  be  definitely  known  also  that  the  greatest  side  pressure 
of  the  cam  against  the  follower  is  at  an  angle  of  40°  as  specified, 
and  that  this  pressure  will  occur  when  HI  of  the  pitch  surface  of  the 
cam  is  at  H,  or  when  the  roller  is  in  contact  with  the  working 
surface  at  H%.  Where  the  cam  form  is  assumed  as  in  Fig.  28,  nothing 
is  known  positively  of  the  starting  and  stopping  velocities  of  the 
follower.  Further,  as  may  be  found  by  trial,  the  maximum  angle  of 
pressure  of  the  cam  against  the  rod  runs  up  to  47°  in  Fig. '  28,  as 
shown  at  D4.  The  minimum  radius  of  the  cam  in  Fig.  28  was  taken 
equal  to  that  in  Fig.  29  for  comparison. 

44.  The  two  previous  problems  have  been  given  as  brief  exercises 
without  going  into  all  the  detail  necessary  to  a  full  understanding, 
in  order  to  give  an  idea  of  the  method  of  producing  cams  on  a  scientific 
basis.     In  the  problems  which  will  follow,  the  several  steps  in  building 
cams  of  various  types  will  be  explained.     In  many  of  the  problems 
the  same  data  will  be  used  so  that  comparisons  of  different  forms 
of  cams  which  produce  the  same  results  may  be  made. 

45.  PROBLEM    3.     SINGLE-STEP   RADIAL    CAM,  PRESSURE   ANGLE 
EQUAL  ON  BOTH  STROKES.      Required  a  single-step  radial  cam  in 
which  the  center  of  the  follower  roller  moves  in  a  radial  line.     The 
maximum  pressure  angle  to  be  30°,  and  the  follower  to  move: 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


29 


\ 


(a)  Up  3  units  in  90°  with  uniform  acceleration 
and  retardation. 

(b)  Down     3     units     in     90°    with    uniform 
acceleration    and    retardation. 

(c)  At  rest  for  180° 

46.  The  first  step  in  the  solution  is  to  determine 
the  total  length  of  the  cam  chart  for  a  parabola 
chart  curve  and  for  a  30°  maximum  pressure  angle. 
From  the  table,  paragraph  30,  the  factor  for  this 
case  is  found  to  be  3.46.  Since  the  travel  of  the 
follower  is  3  units  in  %  revolution,  the  total  length 
of  chart  will  be(_3  X  3.46  X  4  =  41.52,  which, 
therefore,  is  the  length  of  the  chart  A  A'  in  Fig.  30. 
This  length  represents  the  360°  of  the  cam.  -Lay 
off  A  W  equal  to  90°,  according  to  item  (a)  in 
the  data.  Construct  the  parabolic  curve  A  E  C. 
Completing  the  entire  chart,  the  base  curve  is 
found  to  be  A  C  M  N  A'.  The  next  step  is  to  find 
the  radius  of  the  pitch  circle.  The  circumference 
of  this  circle  is  equal  to  the  length  of  the  pitch 

•  <C  41  52 

line  D  Df .    Its  radius  is,  therefore,  equal  to  ~ —  = 

L  TT 

6.61,  and  this  value  is  laid  off  at  0  D,  Fig.  31,  and 
the  pitch  circle  D  F  Q  W  drawn.  The  quadrant 
D  F  is  divided  into  the  same  number  of  parts 
as  D  F  in  Fig.  30.  The  vertical  construction  lines 
H  HI,  II  i,  J  Ji  .  .  .  in  Fig.  30  now  become  the 
radial  lines  correspondingly  lettered  in  Fig.  31, 
and  the  pitch  surface  is  drawn  through  the  points 
AHiIiJi.  .  .  .  The  positions  of  maximum  pres- 
sure are  shown  at  E  and  Q;  at  all  other  points  it 
will  be  less.  The  working  surface  B  G  R  P  is 
found  by  assuming  a  radius  A  B  for  the  roller, 
and  by  striking  a  series  of  arcs  as  shown  at  H2, 
/2,  «/2  •  •  •  with  the  points  Hi,Ii,Ji  ...  as  cen- 
ters, and  then  drawing  the  working  curve  tangent 
to  these  arcs.  With  the  same  specifications  for 
the  up  and  down  motions  of  the  follower,  as 
given  by  items  (a)  and  (b)  in  the  data,  this  type 
of  cam  will  be  symmetrical  about  the  line  Y  C. 


30 


ELEMENTARY   CAMS 


FIG.  31. — PEOBLEM  3,  CAM  LAID  OUT  FROM  CAM  CHART 


FIG.  32. — PROBLEM  3,  CAM  LAID  OUT  INDEPENDENTLY  or  CAM  CHART 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  31 

47.  OMISSION  OF  CAM  CHART.     When  the  relation  between  pres- 
sure angle,  chart  base  and  pitch  lines,  and  cam  pitch  and  surface 
lines  is  understood  and  fixed  in  mind,  the  actual  drawing  of  the 
chart  for  the  graphical  construction  of  simple  cams  and  particularly 
of  single-step  cams  may  be  omitted  with  full  confidence  when  the 
elementary  base  curves  are  used.     For  example,  the  problem  in  the 
previous  paragraph  is  shown   completely  worked  out  in  Fig.  32 
without  any  reference  whatever  to  the  chart  of  Fig.  30.     The  radius 
0  D  of  the  pitch  circle,  Fig.  32,  is  obtained  directly  from  the  formula, 

r  =  57.3  -r-  given  in  paragraph  29.     Substituting  the  data  as  given 

3X3  46 
in  the  previous  paragraph,  r  =  57.3 ^ —  =  6.61  and  is  laid  off 

at  D  0.  The  assigned  motion  of  the  follower  is  laid  off  symmetri- 
cally on  both  sides  of  the  pitch  point  D,  as  at  A  V,  and  the  distances 
A  D  and  V  D  are  divided  into  the  desired  number  of  unequal  parts, 
as  at  1,  4,  9,  16.  The  quadrant  D  F  is  divided  into  the  same  number 
of  equal  parts  as  at  H,  /,  J  .  .  .  and  indefinite  radial  construction 
lines  drawn  through  the  points.  Circular  construction  arcs  are 
next  drawn  through  the  points  1,  4,  9  ...  until  they  intersect  the 
radial  lines,  thus  obtaining  points  HI,  /i,  Ji  ...  on  the  cam  pitch 
surface.  In  general,  a  neater  construction  is  obtained  by  omitting 
the  full  length  of  the  construction  arcs,  as  from  V  to  C  .  .  .  and 
simply  drawing  short  portions  of  the  arc  at  the  intersecting  radial 
lines  as  shown  in  the  lower  left-hand  quadrant  between  C  and  M . 

48.  EXERCISE   PROBLEM  3a.     Required  a  single-step  radial  cam 
in  which  the  center  of  the  follower  roller  moves  in  a  radial  line. 
The  maximum  pressure  angle  to  be  40°,  and  the  follower  to  move: 

(a)  Out  6  units  in  135°  on  the  crank  curve. 

(b)  In    6     "      "  135°  "     "       " 

(c)  At  rest  for  90°. 

49.  PROBLEM  4.    SINGLE-STEP  RADIAL  CAM,  PRESSURE  ANGLES 
UNEQUAL  ON  THE  TWO  STROKES.     Required  a  single-step  radial  cam 
in  which  the  center  of  the  follower  moves  in  a  radial  line.     The 
maximum  pressure  angle  not  to  exceed  30°  on  the  outstroke  nor  50° 
on  the  return  stroke,  and  the  follower  to  move: 

(a)  Out  2  units  in  /i6  revolution  on  the  crank  curve. 

(b)  In    2     "      "  3/6 

(c)  At  rest  for        y%  revolution. 


32 


ELEMENTARY   CAMS 


50.  The  diameter  of  pitch  circle  of  the  cam  that  will  be  necessary 
to  fulfil  the  requirements  on  the  outstroke  will  be : 

2X2  72  X  16 
da  =  — Q  14  y  5 —  =  5.54  units,  or  from  formula  paragraph  29. 

r  =  . 159  2X2'72X16  =  2.77,        '  . 

o 

and  the  diameter  of  pitch  circle  required  for  the  instroke  will  be 

2  X  1.32  X  16 
*=        3.14  X  3 

Inasmuch  as  there  can  be  only  one  pitch  circle  for  a  cam,  the 
largest  one  resulting  from  the  several  specifications  must  be  used. 
In  this  problem  then  the  diameter  S  D  of  the  pitch  circle  in  Fig.  33 


FIG.  33. — PROBLEM  4,  MAXIMUM  PEESSUEE  ANGLE  DIFFERENT  ON  THE  Two  STROKES 

equals  5.54  units.  The  follower's  motion  of  two  units  is  laid  out 
at  A  V  and  the  pitch  surface  A  E  C  M  N  constructed.  The  working 
surface  of  the  cam  B  KG,  etc.,  is  then  drawn.  Since  a  larger  diameter 
of  pitch  circle  had  to  be  used  for  the  return  stroke  than  the  require- 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  33 

ments  called  for,  it  follows  that  the  pressure  angle  will  not  reach 
50°  on  that  stroke,  and  it  may  be  of  some  interest  to  determine  what 
the  maximum  pressure  angle  on  the  return  stroke  will  be.  Sub- 
stituting the  diameter  used,  5.54,  in  the  formula  d  =  —  and  solving 

7T  o 

for  /,  /  is  found  to  be  equal  to  1.63.  From  the  chart  in  Fig.  21  it 
is  shown  that  a  factor  of  1.63  for  the  crank  curve  corresponds  to  a 
maximum  pressure  angle  of  nearly  44°,  and  this  angle  may  be  drawn 
in  its  proper  position  at  Q  in  Fig.  33. 

51.  EXERCISE  PROBLEM  4a.     Required  a  single-step  radial  cam 
in  which  the  center  of  the  follower  roller  moves  in  a  radial  line.     The 
maximum  pressure"  not  to  exceed  30°  on  the  up  stroke  nor  40°  on  the 
down  stroke,  and  the  follower  to  move: 

(a)  Up  3  units  in  135°  on  the  parabola  curve. 

(b)  At  rest  for  45°. 

(c)  Down  3  units  in  90°  on  the  parabola  curve. 

(d)  At  rest  for  90°. 

52.  PRESSURE  ANGLE  INCREASES  AS  PITCH  SIZE  OF  CAM  DECREASES. 
This  is  illustrated  in  Fig.  34,  where  the  large  pitch  cam  represented 
by  D,  D2  .  .  .  gives  exactly  the  same  motion  to  a  follower  as  the 
small  pitch  cam  d,  ck.    .   .   .     It  will  be  noted  that  the  pressure  angle 
for  the  large  cam,  at  the  start,  is  H  D  G,  while  for  the  small  cam  it 
is  increased  to  h  d  g.     Likewise  the  maximum  pressure  angle  for  the 
large  cam,  when  the  follower  is  near  the  end  of  its  stroke,  is  61, 
while  for  the  small  cam  the  maximum  pressure  angle  is  6,  which  is 
larger  than  bi.     From  these  observations  it  may  be  said,  in  general, 
that  the  larger  the  pitch  surface  of  the  cam  the  smaller  will  be  the 
pressure  angle.     The  size  of  the  roller  has  no  effect  whatever  on 
the  pressure  angle.     Two  cams  of  the  same  pitch  size  may  be  of 
totally  different  actual  sizes  for  the  same  work,  one  cam  having 
a  large  roller  and  the  other  a  small  roller.     Therefore  it  is  important 
to  remember  that,  in  general,  the  pressure  angle  may  be  regulated 
by  changing  the  size  of  the  pitch  surface  only  and  not  the  working 
surface. 

53.  CHANGE  OF  PRESSURE  ANGLE  IN  PASSING  FROM  CHART  TO 
CAM.     The  circumference  of  the  pitch  circle  of  the  cam,  it  will  be 
recalled,  is  equal  to  the  length  of  the  pitch  line  on  the  chart.     It 
will  also  be  remembered  that  the  pitch  line  may  be  at  various  heights 
on  the  chart,  paragraph  23.     It  is  now  important  to  consider: 

1st.  That  the  pressure  angle  at  the  pitch  circle  on  the  cam  must 
be  the  same  as  the  pressure  angle  at  the  pitch  line  on  the  chart. 


34 


ELEMENTARY   CAMS 


2d.  That  the  pressure  angle  at  any  point  on  the  pitch  surface 
of  the  cam  outside  of  the  pitch  circle  will  be  less  than  the  pressure 
angle  of  the  corresponding  point  on  the  base  curve  of  the  cam  chart. 

3d.  That  the  pressure  angle  at  any  point  on  the  pitch  surface 


FIG.  34. — SHOWING  KELATION  BETWEEN  PRESSURE  .ANGLE  AND  SIZE  OF  PITCH  CAM 

of  the  cam  inside  of  the  pitch  circle  will  be  greater  than  the  pressure 
angle  of  the  corresponding  point  on  the  base  curve  of  the  cam  chart. 

These  statements,  which  are  theoretically  true  for  nearly  all  cases, 
and  practically  so  for  all  other  cases  where  the  usual  base  curves 
are  employed,  are  demonstrated  in  the  following  paragraph. 

54.  CAM  CONSIDERED  AS  A  BENT  CHART.  Consider  that  the  cam 
itself  is  the  cam  chart  bent  in  its  own  plane  so  that  the  pitch  line 


CAM  PROBLEMS  AND  EXERCISE  PROBLEMS 


35 


becomes  the  pitch  circle.  Then  the  line  D  D',  Fig.  30,  becomes 
the  circle  D  F  0  W,  Fig.  31;  the  line  V  Vr  is  stretched  to  become 
the  circle  V  C  S  Y,  and  the  straight  line  AM  A'  is  compressed  to 
become  the  circle  AM  A.  This  means,  in  a  general  way,  that 
the  rectangle  D  V  Vf  D' ,  Fig.  30,  is  so  distorted  that  if  an  original 
diagonal  had  been  drawn  from  D  to  Vf  it  would  have  an  increased 
length  and  a  decreasing  slant  after  the  bending  had  taken  place. 
With  a  decreasing  slant  of  the  pitch  surface  the  pressure  angle  will 
decrease.  Likewise,  a  diagonal  drawn  from  D'  to  A  in  the  original 
rectangular  chart  would  be  decreased  in  length  and  would  have  an 
increasing  slant,  and  the  pressure  angle  would  be  increasing  toward 
A.  This  is  illustrated  in  detail  in  Figs.  35  and  36. 

55.  BASE  LINE  ANGLES,  BEFORE  AND  AFTER  BENDING.     The  pres- 
sure angle  of  30°  at  E  in  Fig.  35  is  reduced  to  23°  in  Fig.  36,  and  the 


Fia.  35. — SECTION  OF  CAM  CHART  BE-        FIG.  36. — SECTION   OF    CAM    CHART   AFTER 
FORE  BENDING  BENDING,  BC  CONSTANT  IN  BOTH  FIGURES 

30°  at  D  are  increased  to  41°.  Fig.  35  represents  a  cam  chart  with 
a  straight  base  line  D  E,  and  Fig.  36  is  a  corresponding  cam  sector 
with  D  E  as  the  pitch  surface.  If  B  C,  Fig.  35,  is  taken  as  the 
pitch  line,  B  C,  Fig.  36,  will  be  part  of  the  pitch  circle.  The  uniform 
pressure  angle  of  30°  from  A  to  E,  Fig.  35,  will  grow  smaller  beyond 
A  in  Fig.  36  for  the  reason  that  the  radial  components  of  the  tan- 
gential triangles  remain  constant,  as  illustrated  at  L  M,  while  the 
tangential  components  grow  longer  as  illustrated  from  A  N  to  E  L, 
which  are  respectively  equal  to  the  arcs  A  Y  and  E  LI.  Con- 
sequently, the  angles  grow  smaller  from  the  angle  N  A  P  to  L  E  M. 
Similarly  it  may  be  shown  that  they  grow  larger  from  N  A  P  to 
QDR. 

56.  LIMITING  SIZE  OF  FOLLOWER  ROLLER.     The  radius  of  the 
follower  roller  may  be  equal  to,  but  in  general  should  be  less  than 


36 


ELEMENTARY   CAMS 


the  shortest  radius  of  curvature  of  the  pitch  surface,  when  measured 
on  the  working-surface  side.  If  the  radius  of  the  roller  is  not  so 
taken,  the  follower,  when  put  in  service,  will  not  have  the  motion 
for  which  it  was  designed. 

57.  CASE  1.     RADIUS  OF  ROLLER  EQUAL  TO  RADIUS  OF  CURVATURE 
OF  PITCH  CAM.     In  Fig.  37,  A  B  E  F  A  is  the  pitch  surface  of  a  cam. 


-4- 


FIG.  37. — LIMITING  SIZE  OF  FOLLOWER  ROLLEB 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  37 

G  A  is  the  radius  of  curvature  at  A  and  A  G  is  the  radius  of  the 
roller.  In  this  case  both  radii  are  equal  and  the  working  surface 
has  a  sharp  edge  at  G. 

58.  CASE   II.     RADIUS  OF  ROLLER  GREATER  THAN  RADIUS   OF 
CURVATURE  OF  PITCH  CAM.     From  B  to  C,  Fig.  37,  the  radius  of 
curvature  of  the  pitch  surface  is  H  B,  which  is  less  than  the  roller 
radius.     In  this  case  the  working  surface  will  be  undercut  at  /  in 
generating  the  cam,  and  if  the  cam  is  built  the  center  of  the  roller 
will  mark  the  path  BI  Ji  Ci  instead  of  BiJCi,  and  the  follower  will 
fail  to  move  the  desired  distance  by  the  amount  J  Ji. 

59.  SPECIAL  APPLICATION  OF  CASE  II.     EFFECT  OF  AN  ANGLE  IN 
THE  PITCH  SURFACE  OUTLINE.     This  is  illustrated  at  R  F  Q  in  Fig.  37, 
and  is  a  special  application  of  Case  II,  in  which  the  radius  of  cur- 
vature of  the  cam's  pitch  surface  is  reduced  to  zero.     Undercutting 
is  here  illustrated  by  considering  that  a  cutter,  represented  by  the 
dash  circular  arc,  is  moving  with  its  center  on  the  pitch  surface 
arc  E  F.     It  then  cuts  the  working  surface  M  S.     As  the  center  of 
the  cutter  is  moved  from  F  toward  A,  the  part  W S  of  the  working 
surface  which  was  previously  formed   is   now   cut   away,   leaving 
the  sharp  edge  W  on  which  the  follower  roller  will  turn  when  the 
cam  is  placed  in  operation.     The  center  of  the  follower  roller  will 
then  move  in  the  path  R  T  Q  instead  of  R  F  Q,  and  the  follower  will 
fall  short  of  the  desired  motion  by  the  amount  T  F. 

60.  CASE  III.     RADIUS  OF  ROLLER  LESS  THAN  RADIUS  OF  CURVA- 
TURE OF  PITCH  CAM.     From  D  to  E,  Fig.  37,  the  radius  of  curvature 
of  the  pitch  surface  is  K  D,  which  is  greater  than  the  roller  radius. 
In  this  case,  which  is  the  practical  one,  although  close  to  the  limit, 
a  smooth  curved  working  surface  is  provided  for  the  roller  from 
LtoM. 

61.  RADIUS  OF  ROLLER  NOT  AFFECTED  BY  RADIUS  OF  CURVATURE 
ON  NON-WORKING  SIDE.     From  Ci  to  D,  Fig.  37,  the  radius  of  curva- 
ture of  the  pitch  surface  is  less  than  the  radius  of  the  roller,  but 
this  short  radius  is  not  on  the  working  side  of  the  pitch  surface, 
and  therefore  the  roller  will  roll  on  the  surface  I  L  while  its  center 
travels  on  the  pitch  curve  Ci  D. 

62.  ROLLERS  FOR  POSITIVE-DRIVE  CAMS.     When  the  largest  roller 
for  a  positive  or  double-acting  cam  is  being  determined  the  radius 
of  curvature  on  both  sides  the  pitch-surface  curve  must  be  con- 
sidered and  the  smallest  radius  used.     For  example,  in  Fig.  37,  if 
A  J  E  T  A  were  the  pitch  surface  for  a  double-acting  cam,  N  C 
would    be    the   maximum    roller   radius,    whereas   H  J    would  be 


38 


ELEMENTARY   CAMS 


the   maximum    radius    if   it    were   for    an    external    single-acting 
cam. 

63.  RADIUS  OF  CURVATURE  OF  NON-CIRCULAR  ARCS.  In  illustrat- 
ing the  above  cases  the  pitch  surface  was  assumed  as  being  made 
up  of  straight  lines  and  arcs  of  circles  in  order  to  show  more  effec- 
tively and  more  simply  the  limits  of  action  in  each  instance.  Where 
the  pitch  surface  contains  curves  of  constantly  varying  curvature, 
and  they  generally  do  in  practice,  the  shortest  radius  of  curvature 
of  the  pitch  surface  may  be  found  with  all  necessary  accuracy  by 
trial  with  the  compass,  using  finally  that  radius  whose  circular  arc 
agrees  for  a  small  distance  with  the  irregularly  curved  arc.  For 
example,  in  Fig.  38,  let  G  H  D  J  B  be  a  portion  of  a  pitch  surface 


FIG.  38. — LIMITING  SIZE  OF  FOLLOWER  ROLLER  WORKING  ON  NON-CIRCULAR  CAM  CURVES 

made  up  of  non-circular  arcs.  The  shortest  radius  of  curvature 
on  both  sides  is  found,  by  trial,  to  be  F  H.  The  center  F  is  marked 
and  the  osculatory  arc  X  H  Z  drawn  in.  Then  H  F  is  the  largest 
possible  radius  of  roller  for  a  double-acting  cam,  and  with  this 
roller  the  working  surfaces  will  be  V  F  T  W  and  Vl  FI  T±  Wi. 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  39 

If  a  larger  roller  is  used,  with  a  radius  D  R,  for  example,  the 
working  surfaces  of  the  groove  will  be  S  0  E  and  PI  K\  NI,  and  the 
new  pitch  surface,  after  cutting  the  cam,  will  be  G  C  D  L  B,  if  the 
roller  is  kept  always  in  contact  with  the  inner  surface  of  the  groove. 
If  it  is  kept  always  in  contact  with  the  outer  surface  of  the  groove, 
the  original  pitch  surface  will  be  changed  to  G  C  H  DI  J  B.  In 
either  case  the  original  desired  follower  motion  is  not  obtained  if  the 
roller  is  too  large,  and  if  a  positive-drive  cam  is  run  with  the  larger 
roller  the  follower's  motion  will  be  indeterminate,  the  center  of 
the  roller  having  any  possible  position  between  C  D  L  and  C  H  J  L. 

64.  PROBLEM  5.     DOUBLE-STEP  RADIAL  CAM.     Required  a  double- 
step  radial  cam  in  which  the  center  of  the  follower  roller  moves  in 
a  radial  line.     The  maximum  pressure  angle  to  be  30°,  and  the 
follower  to  move: 

(a)  Up  4  units  in  }/%  revolution  on  the  crank  curve. 

(b)  At  rest  for  ^4  revolution. 

(c)  Up  4  units  in  J/£  revolution  on  the  parabola  curve. 

(d)  Down  2  units  in  %  revolution  on  the  elliptical  curve. 

(e)  At  rest  for  %  revolution. 

(f)  Down  6  units  in  %  revolution  on  the  parabola  curve. 

65.  In  Problem  3  there  are  only  two  motion  assignments,  (a)  and 
(b),  in  the  data,  and  they  were  the  same  except  for  direction.     Con- 
sequently only  one  computation  was  necessary.     When  two  or  more 
dissimilar  assignments  are  made  in  the  data,  as  in  the  present  problem, 
it  is  advisable  to  make  a  computation  for  the  length  of  the  chart 
diagram  for  each  motion  specification,  as  follows: 

(a)  4  X  2.72  X  8  =    87.04,  which  is  the  length  of  chart  and  of 

the   pitch    circle   circumference  = 
13.86  pitch  circle  radius. 

(c)  4  X  3.46  X  8  =  110.72,  which  is  the  length  of  chart  and  pitch 

circle  circumference  =  17.62  pitch 
circle  radius. 

(d)  2  X  3.95  X  8  =    63.20,  which  is  the  length  of  chart  and  pitch 

circle  circumference  =  10.06  pitch 
circle  radius. 

(f)    6  X  3.46  X  4  =    83.04,  which  is  the  length  of  chart  and  pitch 

circle  circumference  =  13.22  pitch 
circle  radius. 

Inasmuch  as  there  is  a  different  length  of  chart  and  a  different 
pitch  line  for  each  item  in  the  data  one  can  not  tell  which  pitch  line 
to  take  without  some  preliminary  computation.  For  this  purpose 


40 


ELEMENTARY   CAMS 


a  chart  diagram  is  well  adapted,  as  follows:  Construct  a  rectangle, 
Fig.  39,  with  a  height  A  T  equal  to  the  total  motion  of  the  follower 
in  one  direction,  8  units  in  this  case.  Make  the  length  A  A'  of 
rectangle  any  convenient  value  entirely  independent  of  any  of  the 
values  computed  above  and  label  this  according  to  the  longest  chart 
length  as  computed  above.  Lay  off  straight  lines  to  represent  the 
component  parts  of  the  base  curve  as  assigned  in  the  data  and  label 
them  as  shown  at  A  C,  C  B,  B  H,  etc.  Draw  the  several  pitch  lines 
as  at  F  D,  J  I,  etc. 

66.  For  general  procedure,  consider  the  pitch  line  which  passes 
through  the  point  calling  for  the  longest  chart  length.  This  will 
be  the  pitch  line  J  I  passing  through  G,  Fig.  39,  which  calls  for  a 
chart  length  of  110.72  and  a  pitch  radius  of  17.62.  If  G  is  to  be  at 


M        NL 

^7 

SE 

m 

4 

1 

±c5 

w 

6 

,/ 

3/,t      L 

°%\ 

B 

, 

,  * 

C               4     < 

] 

t 

i 

J1 

\     *D 

z 

P 

i 

J 

£\  r 

<A                %  110  7°     *                *                %                * 

X 

FIG.  39. — PROBLEM  5,  CAM  CHART  DIAGRAM  FOR  DOUBLE-STEP  CAM 

a  radius  of  17.62  in  the  cam,  E  will  be  at  a  radius  of  17.62  —  4  = 
13.62.  But  from  computation  (a)  it  is  seen  that  E  must  be  at  a 
radius  of  13.86.  Therefore,  if  the  radius  of  cam  pitch  circle  is 
retained  at  17.62,  the  trial  pitch  line,  J  7,  on  the  chart  diagram  will 
have  to  be  lowered,  13.86  —  13.62  =  .24,  giving  the  new  pitch  line 
U  U'.  If  the  line  U  Uf  now  becomes  the  pitch  circle  the  point  E 
will  be  at  17.62  —  3.76  =  13.86,  just  as  called  for  in  computation 
(a),  and  the  pressure  angle  will  be  30°  at  the  point  E  on  the  cam. 

The  other  critical  points  at  P  and  K  must  also  be  tested  with 
respect  to  the  proposed  pitch  line,  U  U'.  With  this  pitch  line  the 
point  P  will  be  2.76  inside  of  the  pitch  circle,  or  at  a  radius  of  17.62  — 
2.76  =  14.86.  This  is  safe,  as  the  computed  radius  for  P  was  only 
13.22  according  to  item  (f).  The  point  K  is  also  safe,  for  it  will  be 
at  a  radius  of  17.62  +  1.24  =  17.86,  whereas  a  radius  of  only  10.06 
is  required. 

67.  The  cam  may  now  be  drawn  by  constructing  the  true  cam 
chart  as  in  Fig.  40,  which  is  lettered  the  same  as  Fig.  39,  and  plotting 
the  cam  from  it  as  in  Fig.  41.  The  pitch  line  U  U'  of  Fig.  40  be- 


CAM  PROBLEMS  AND  EXERCISE  PROBLEMS 


41 


comes  the  pitch  circle,  having  a  radius  0  U  in 
Fig.  41,  and  the  ordinates  of  Fig.  40  become 
the  radial  measuring  lines  in  Fig.  41.  Or  the 
cam  may  be  drawn  directly,  without  the  use  of  a 
cam  chart,  as  indicated  in  Fig.  42,  where  the 
pitch  circle  0  U'  is  first  drawn  with  a  radius  of 
17.62.  The  assigned  angles  are  then  laid  down 
and  the  several  pitch  curves,  such  as  A'  E  C,  are 
constructed  at  the  proper  radial  distances  as  de- 
termined in  Fig.  39  and  as  illustrated  for  one 
case  at  E  Ei  (3.76)  in  Fig.  42. 

68.  DETERMINATION  OF  MAXIMUM  PRESSURE 
ANGLE  FOR  EACH  OF  THE  CURVES  MAKING  UP  A 
MULTIPLE-STEP  CAM.  If  it  is  desired  to  know  the 
exact  pressure  angle  at  P,  Fig.  41,  it  may  be  readily 
determined  by  making  the  value  of  r  =  (17.62  — 

hf 


2.76)=  14.86  in  the  formula,  r  =  .159 


and 


solving  for  /,  the  notation  being  the  same  as  given 
in  paragraph  29. 

14.86 


.159  X  6  X  4 


=  3.89. 


Consulting  the  chart  of  cam  factors  in  Fig.  21, 
it  is  found  that  a  factor  of  3.89,  when  applied  to 
the  parabola  chart  curve,  shows  a  cam  pressure 
angle  of  about  27°,  which  is  under  the  assigned 
limit,  and  therefore  need  not  be  further  consid- 
ered. In  a  similar  manner  the  pressure  angle  at 
G  and  K  on  the  cam  may  be  computed  if  desired, 
the  former  being  a  small  fraction  of  a  degree 
under  30°  and  the  latter  something  less  than  20°, 
the  reading  running  off  the  chart. 

69.  EXERCISE  PROBLEM  5a.  Required  a  double- 
step  radial  periphery  cam  in  which  the  center  of 
the  follower  roller  moves  in  a  radial  line.  The 
maximum  pressure  angle  to  be  30°  and  the  follower 
to  move: 

(a)  Out  5  units  in  150°,  with  uniform  accelera- 
tion and  retardation. 

(b)  In  2  units  in  45°  on  the  crank  curve. 


42  ELEMENTARY   CAMS 

(c)  At  rest  for  45°. 

(d)  In  3  units  in  120°  on  the  crank  curve. 

70.  PROBLEM   6.     CAM  WITH  OFFSET  ROLLER  FOLLOWER.     Re- 
quired a  single-step  radial  periphery  cam  in  which  the  center  of  the 
follower  roller  moves  forth  and  back  in  a  straight  line  which  does 
not  pass  through  the  center  of  rotation  of  the  cam.     The  maximum 
pressure  angle  when  the  follower  is  at  the  bottom  of  its  stroke  is 
to  be  30°,  and  the  follower  is  to  move: 

(a)  Up       3  units  in  90°  on  the  parabola  curve. 

(b)  Down  3     "      "  90°   "     " 

(c)  At  rest  for  180°. 

71.  Problems  of  this  nature  are  totally  different,  both  in  pressure- 
angle  action  and  in  methods  of  construction,  from  the  preceding 
ones.     As  may  be  noted  in  the  data,  it  is  required  that  the  pressure 
angle,  when  the  follower  is  at  rest  at  the  bottom  of  its  stroke,  shall  be  30°. 
It  will  appear  presently  that  the  pressure  angle,  when  the  follower 
is  in  motion,  may  be  zero  or  even  negative  on  one  of  the  strokes  in 
this  form  of  cam.     It  will  also  be  shown  that  the  maximum  pressure 
angle  during  the  follower  motion  cannot  be  assigned  in  advance  and 
obtained  in  any  practical  manner.     From  the  above  it  follows  that 
the  offset  radial  cam  has  a  peculiar  advantage  in  keeping  considerable 
side  pressure  off  the  follower  guides  during  the  time  that  the  follower 
is  moving  in  one  direction,  although  at  the  bottom  of  the  stroke  the 
pressure  angle  may  have  any  desired  value,  and  during  the  period 
of  motion  in  the  opposite  direction  the  pressure  angle  will  reach  a 
maximum  value  much  larger  than  the  assigned  angle  at  the  bottom 
of  the  stroke. 

72.  The  method  of  construction  for  the  offset  roller  cam  is  illus- 
trated in  Fig.  43.     The  diameter  of  the  pitch  circle,  U  Fr  S  T,  is  com- 
puted as  before  by  the  formula,  d  =  114.6  -/,  and  found  to  be  13.22 

units.  An  angle  equal  to  the  assigned  pressure  angle  is  then  laid 
off  at  U  0  U',  U  0  being  parallel  to  the  direction  of  motion  of  the 
follower.  Draw  a  line,  DW,  parallel  to  U  0  and  so  located  that  it 
has  an  intercept  D  A  between  the  pitch  circle  and  the  inclined  line 
equal  to  one-half  the  travel  of  the  follower.  This  may  be  done  by 
trial,  or  graphically,  as  shown  by  the  dotted-line  construction  which 
is  drawn  at  Yr  X  A'  instead  of  at  Y  to  avoid  complication  of  con- 
struction lines.  The  angle  U  0  Y'  equals  the  angle  U  0  Y.  Y'  X, 
parallel  to  U  0,  is  drawn  equal  to  one-half  the  stroke.  An  arc  X  Af  % 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


43 


parallel  to  the  arc  Yf  U,  is  drawn  through  X  by  using  U  0  as  a  radius 
and  Z  as  a  center,  where  0  Z  equals  Yf  X.  A  circular  arc  through 
A'  with  0  as  a  center  will  intersect  O  Uf  in  the  desired  point  A.  The 
point  A  will  then  be  the  lowest  point  of  the  stroke,  D  will  be  the 
center  of  the  stroke,  and  W  0  the  radius  of  the  construction  circle. 


H  '•> 


FIG.  41. — PROBLEM  5,  DOUBLE-STEP  CAM  CONSTRUCTED  FROM  CAM  CHART 


FIG.  42. — PROBLEM  5,  DOUBLE-STEP  CAM  CONSTRUCTED  WITHOUT  USE  OF  CAM  CHART 

The  distance  A  V  is  equal  to  the  assigned  3  units  of  motion,  and 
the  divisions  1,  4,  9  .  .  .  are  made  according  to  the  requirements  of 
the  parabola  curve.  The  assigned  90°  is  laid  off  on  the  construction 
circle  at  W  F  and  divided  into  a  number  of  equal  arcs  at  H,  I, 
J  .  .  .  corresponding  to  the  number  of  divisions  at  A  V,  eight  being 
used  in  the  present  example.  Tangents  to  the  construction  circle, 


44  ELEMENTARY   CAMS 

such  as  H  HI,  I  1 1,  J  Ji  .  .  .  are  then  drawn  at  H,  7,  J  .  .  .  and  the 
distances  Wl,  W4,  W9  .  .  .  laid  off  on  these  tangents,  thus  giving 
the  points  HI,  I\y  Ji  .  .  .  on  the  pitch  surface  of  the  cam.  Or  these 
latter  points  may  be  obtained  by  swinging  arcs  through  1,  4,  9  .  .  . 
about  0  as  a  center,  until  they  meet  the  respective  tangents  at  HI, 

7l,    /*:... 

73.  An  examination  of  the  pressure  angles  for  a  cam  with- an 
offset  follower  shows  that  during  the  up  stroke  the  pressure  angles 
are  very  small,  being,  in  fact,  negative  from  Ji  to  KI,  Fig.  43,  and, 
when  measured,  the  average  pressure  angle  for^he  working  or  up 
stroke  is  between  6  and  7  degrees  in  this  problem;   although  on  the 
down  or  return  stroke  it  reaches  an  average  of  between  37°  and  38° 
and  a  maximum  of  46°  near  Qf .     In  this  class  of  problem  the  com- 
putation for  diameter  of  pitch  circle  serves  merely  as  a  guide  in  deter- 
mining a  size  that  will  give  a  small  cam  and  a  small  average  pressure 
angle  on  the  working  stroke.     If  the  diameter  of  the  pitch  circle  is 
arbitrarily  taken  either  larger  or  smaller  than  the  value,  as  above 
computed,  or  if  other  base  curves  are  used,  the  negative  pressure 
angles  at  Ji,  EI,  and  KI  may  disappear  entirely;  which  would  be  an 
advantage  where  it  is  desired  to  have  pressure  on  the  follower  guides 
on  one  side  only. 

74.  It  has  doubtless  been  observed  that  there  is  a  decided  lack 
of  symmetry  in  this  form  of  cam,  even  though  the  data  are  similar 
for  both  strokes  of  the  follower.     This  is  illustrated  in  Fig.  43,  where 
the  portion  A  C  of  the  pitch  surface  for  the  outstroke  is  quite  different 
from  the  portion  C  M.     It  is  also  characteristic  of  this  form  of  cam 
that  the  pitch  and  working  curves  each  embrace  either  a  smaller  or 
a  larger  angle  than  the  assigned  angle  for  a  given  stroke  of  the  follower, 
as  shown  by  the  angle  A  0  C  being  less,  and  the  angle  COM  being 
greater,  than  the  assigned  90°.     This,  of  course,  is  due  to  the  fact 
that  when  C  has  traveled  90°  to  V  the  line  0  C  will  have  passed  the 
original  zero  line  0  A  of  the  pitch  curve  and  will  be  in  the  position* 
0  V.     Therefore,  the  cam  angle  for  one  stroke  of  the  follower  will  be 
less  than  the  assigned  angle  by  the  amount  of  the  angle  included  by 
V  0  A;  for  the  other  stroke  it  will  be  greater  than  the  assigned 
angle  by  the  same  amount. 

75.  EXERCISE  PROBLEM  6a.     Required  a  single-step  radial  periph- 
ery cam  in  which  the  center  of  the  follower  roller  moves  forth  and 
back  in  a  straight  line  which  does  not  pass  through  the  center  of 
rotation  of  the  cam.     The  maximum  pressure  angle  when  the  follower 
is  at  the  bottom  of  its  stroke  is  to  be  30°,  and  the  follower  is  to  move: 


CAM  PROBLEMS  AND  EXERCISE  PROBLEMS 


45 


(a)  Out  6  units  in  135°  on  the  crank  curve. 

(b)  Jn     6     "     "   135°   "    "       " 

(c)  Rest  for  90°. 

In  this  problem,  only  the  initial  pressure  angle  at  the  bottom  of 
the  stroke  need  be  shown;  th.e  pressure  angles  at  other  positions, 
such  as  are  shown  in  Fig.  43  at  HI,  I\,  may  be  omitted. 


u1 


FIG.  43. — PROBLEM  6,  CAM  WITH  OFFSET  ROLLER  FOLLOWER 

76.  PROBLEM  7.     CAM  WITH  FLAT  SURFACE  FOLLOWER, — MUSH- 
ROOM CAM.     Required  a  radial  periphery  cam  to  operate  an  offset 
follower  which  has  a  flat  surface  instead  of  a  roller.     The  follower 
to  move: 

(a)  Up       3  units  in  90°  on  the  parabola  base. 

(b)  Down  3     "      "  90°  "     " 

(c)  At  rest  for  180°. 

77.  This  type  of  cam  is  known  also  as  the  mushroom  cam.     Flat 


ELEMENTARY  CAMS 


surface  followers  may  be  offset  as  shown  in  the  side  and  top  views 
in  Fig.  44,  where  the  center  line  N"  Y"  of  the  follower  spindle  is  set 
the  distance  P"  0"  in  front  of  the  center  of  the  cam  plate.  In  this 
case  there  will  be  a  part  sliding  and  part  rolling,  of  the  cam  on  the 
follower  and  the  follower  will  turn  about  its  own  axis,  N"  Y"  as  it 


FIG.  44. — PROBLEM  7,  CAM  WITH  FLAT  SURFACE  FOLLOWER— MUSHROOM  CAM 

is  being  raised  and  lowered.  When  the  follower  is  not  offset,  i.e., 
when  the  center  line  0"  N"  is  placed  in  line  with  M "  P",  the  action 
will  be  all  sliding  and  there  will  be  no  turning  of  the  follower  spindle 
on  its  axis.  In  this  case  there  will  be  localized  wear  on  the  follower, 
while  in  the  former  case  the  wear  will  be  more  widely  distributed 
over  the  follower  surface.  In  both  cases  the  construction  is  the 
same  and  is  explained  in  the  following  paragraph. 

78.  In  cam  followers  having  flat  surfaces  perpendicular  to  the 
line  of  action,  the  line  of  pressure  is  M"  Q"  and  is  parallel  to  the 
line  of  action  of  the  follower,  instead  of  being  inclined  to  it  as  in 
the  case  of  cams  having  roller  followers.  Because  of  this  character- 
istic action  the  ordinary  pressure-angle  factors  do  not  apply  in 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  47 

cams  of  this  class  in  computing  or  obtaining  the  diameter  of  the 
pitch  circle  D  F  S  T,  and  this  circle  may  be  assumed.  In  some 
cases  a  fair  guide  for  the  size  of  this  circle  may  be  obtained  by  using 

the  regular  formula,  d  =  114.6-r-,  for  diameter  of  pitch  circle,  as- 
suming the  30°  pressure  angle  factor.  Solving,  r  is  found  to  equal 
6.61,  and  is  laid  off  at  0  D.  The  assigned  three  units  of  motion  are 
then  laid  off,  one-half  on  each  side  of  D,  as  at  A  and  V.  The  assigned 
90°  are  next  laid  off  at  A  0  F  and  divided  into  the  desired  number 
of  construction  parts,  four  being  used  in  this  case,  as  at  0  D\y  0  D2j 
.  .  .  The  distance  A  F  is  also  divided  into  four  parts,  A  H  and  V  K 
being  each  equal  to  1  unit  and  H  D  and  K  D  equal  to  3  units.  Only 
four  divisions  are  taken  in  this  case  to  avoid  confusion  of  lines  in 
the  illustration,  but  in  student  problems  6  or  8  points  should  be 
taken,  and  in  practical  work  12  to  24  divisions  should  be  used. 
The  first  division  point,  H,  is  now  revolved  to  meet  the  first  radial 
division  line  0  DI,  thus  giving  the  point  HI,  where  a  line  HI  E  is 
drawn  perpendicular  to  HI  0.  This  line  HI  E  represents  the  bottom 
of  the  follower  disk  A  C  with  reference  to  the  cam  when  the  cam 
has  turned  through  the  angle  A  0  HI.  The  points  D2  and  KI  are 
obtained  in  the  same  manner  as  was  HI  and  corresponding  perpendic- 
ulars are  drawn,  as  at  D2  D*  and  KI  K2.  As  smooth  a  curve  as  possible 
is  now  drawn  tangent  to  these  perpendiculars  and  the  points  of 
tangency  marked  as  at  H2}  Z>4,  and  K2.  This  smooth  curve,  A  G, 
is  the  working  surface  of  the  cam. 

79.  The  size  of  the  follower  must  also  be  determined.  The  most 
satisfactory  way. of  doing  this  is  to  find,  first,  the  locus,  or  path, 
of  the  line  of  contact  between  the  periphery  of  the  cam  and  the 
follower  disk.  This  is  obtained  by  considering  that  when  HI  is  at 
H,  the  point  of  tangency  H2  is  at  H$,  the  length  H  Hs  being  equal  to 
HI  H2.  Likewise,  when  D2  is  at  D,  D4  is  at  Z)5,  and  the  same  for  the 
other  points  of  tangency.  The  dash  line  curve  through  the  points 
A  H3  D6  K3  .  .  .  is  the  locus  of  contact,  between  the  cam  and  the 
follower.  The  point  L  is  the  extreme  point  of  this  curve  and  if  the 
follower  were  not  offset,  the  length  of  an  ordinary  toe  or  flat  extension 
of  the  follower  would  have  to  be  at  least  equal  to  Nr  X.r  If  the 
follower  is  offset,  say  by  the  amount  Nf  Mr  (=  N"  M"),  the  radius 
of  the  disk  will  have  to  be  at  least  equal  to  Nf  Z/,  and  the  extreme 
line  of  contact  will  be  L'  Jf .  The  other  extreme  line  of  contact  will 
be  a  similar  line  through  L"7,  and  the  area  of  the  flat  disk  which  will 


48 


ELEMENTARY   CAMS 


be  subject  to  wear  will  be  the  annular  surface  between  the  periphery 
and  the  dashline  circle  whose  radius  is  Nf  A '.  As  to  the  wear  on  the 
cam  itself,  there  would  be  pure  sliding  of  the  curved  surface  A  G 
on  the  flat  surface  A  X  if  the  follower  were  not  offset.  With  an 
offset  follower  there  is  an  effective  turning  radius  equal  to  the  offset 


FIG.  44. — (Duplicate).    PROBLEM  7,  CAM  WITH  FLAT  SURFACE  FOLLOWER — 
MUSHROOM  CAM 

•Nf  M'  tending  to  rotate  the  follower  about  its  axis  N"  F/r,'and  this 
changes  the  action  of  the  cam  on  the  follower  entirely  by  causing 
part  rolling  and  part  sliding. 

80.  The  pressure  angle  in  this  form  of  cam  must  be  considered 
differently  from  cams  which  operate  against  rollers.  In  roller 
follower  cams  it  is  the  angle  between  the  normal  to  the  cam  surface 
and  the  line  of  action  of  the  follower  that  determines  the  side  pres- 
sure on  the  bearings,  whereas  in  flat  surface  followers  it  is  the  distance 
that  the  line  of  contact  is  away  from  the  line  of  action  that  determines 
it.  This  distance  varies  constantly,  and  in  the  illustration  in  Fig. 
44  the  limits  of  variation  are  M'  N'  and  Qf  N'.  These  are,  in  reality, 
lever  arms  on  which  the  pressure  acts  to  produce  a  turning  moment 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  49 

which  must  be  resisted  by  the  follower  guides.  Since  there  can  be 
no  pure  rolling  action  between  the  cam  and  follower  in  constructions 
of  this  type,  there  is  nothing  to  be  gained  in  this  particular  by  a 
large  offset.  On  the  contrary,  there  is  much  to  be  lost,  due  to  the 
large  turning  moment  on  the  follower  rod.  A  fair  guide  as  to  the 
offset  would  be  to  keep  the  angle  formed  by  the  center  line  Y"  0" 
of  the  follower  motion  and  the  line  Y"  M"  or  Y"  Q"  joining  the 
center  of  the  bearing  with  the  midpoint  of  the  line  of  contact,  to 
within,  say,  30°,  or  any  other  maximum  value  that  circumstance 
might  warrant.  The  angle  here  defined  might  be  termed  the  pres- 
sure angle  in  this  type  of  cam.  The  minimum  pressure  angle, 
N"  Y"  M",  is  seen  in  its  true  size,  while  the  maximum  pressure 
angle  as  projected  at  U"  Y"  Q"  must  be  revolved  about  U"  Y" 
as  an  axis  until  U"  Q"  equals  Nr  Q',  when  it  will  appear  in  its  true 
size  as  at  U"  Y"  Q'". 

81.  LIMITED  USE  OF  CAMS  WITH  FLAT  SURFACE  FOLLOWERS.     Cams 
with  followers  of  this  type  are  not  well  adapted,  in  general,  for  cases 
in  which  the  follower  must  have  specified  velocities  during  its  stroke. 
If  the  follower  is  required  only  to  move  from  one  end  of  its  stroke 
to  the  other  in  a  given  period  of  time,  independently  of  all  inter- 
mediate velocities,  this  form  of  construction  may  be  readily  applied. 
The  principal  difficulties  to  be  met  in  the  building  of  these  cams, 
when  the  intermediate  velocities  are  specified,  are,  first,  the  large 
time  angles  necessary  for  a  desired  follower  motion,  or,  second,  a 
comparatively  large  cam.     The  cause  of  these  difficulties  may  be 
pointed  out  in  Fig.  44,  where  it  may  be  seen  that  the  construction 
point,  KI,  might  have  been  so  much  further  out  radially  that  the 
perpendicular  line,  KI  K2,  would  have  passed  to  the  left  of  R  and  it 
would  have  been  impossible  to  draw  the  smooth  cam  curve  A  G 
tangent  successively  to  all  the  perpendiculars.     The  limiting  prac- 
tical  case   appears  when  any  three  successive   construction  lines 
meet  in  a  point,  in  which  event  the  cam  will  have  a  sharp  edge  and 
be  subject  to  excessive  wear  at  that  point.     This  subject  is  further 
considered  in  paragraph  106. 

82.  If  one  is  not  limited  in  the  time,  or  angle,  in  which  the  follower 
must  do  its  work;  or,  if  not  limited  in  the  size  of  the  cam,  this  form 
of  construction  may  be  used  for  any  set  of  velocity  values  so  long 
as  they  produce  a  working  surface  which  always  curves  outward 
or  which  has  an  edge  which  points  outward. 

83.  EXERCISE  PROBLEM  7a.     Required  a  radial  periphery  cam 
to  operate  -an  offset  follower  which   has   a   flat   surface   perpen- 


50  ELEMENTARY   CAMS 

dicular  to  the   line   of    motion    instead    of    a    roller,    the   follower 
to  move: 

(a)  Up       3  units  in  90°  on  the  crank  curve. 

(b)  Down  3     "      "  90°   "     " 

(c)  Rest  for  180°. 

Take  cam  disk  to  be  one  unit  thick  and  the  follower  offset  equal 
to  two  units  measured  from  center  of  cam  disk.  Find  and  mark 
the  locus  of  contact,  also  the  size  of  the  follower  disk  and  the  area 
of  follower  surface  subject  to  wear. 

84.  CAMS  FOB  SWINGING  FOLLOWER  ARMS.     In  the  previous  prob- 
lems the  motion  of  the  center  of  the  follower  roller  has  been  in  a 
straight  line.     When  the  center  of  the  roller  moves  in  a  curve  a 
different  method  of  construction  is  used  to  advantage.     Cams  with 
swinging  followers  are  illustrated  in  Figs.  45  and  46,  the  arc  of 
swing  A  V  of  the  follower  having  its  extremities  on  a  radial  line  in 
the  former  illustration;    and  on  an  arc  which,   continued,   passes 
through  the  center  of  the  cam  in  the  latter  illustration.     These  two 
forms  of  construction,  although  apparently  differing  in  only  a  slight 
detail,  give  quite  different  results  and  each  has  its  own  particular 
field  of  usefulness.     A  comparison  of  the  results  will  be  given  in 
paragraph  95  after  a  problem  in  each  case  has  been  worked  out. 

85.  PROBLEM  8.     CAM  WITH  SWINGING  FOLLOWER  ARM,  ROLLER 
CONTACT — EXTREMITIES  OF  SWINGING  ARC  ON  RADIAL   LINE.     Re- 
quired a  radial  periphery  cam  to  operate  a  roller  follower  where  the 
follower  arm  swings  about  a  pivot  and  where  the  two  extreme  posi- 
tions of  the  center  of  the  roller  lie  on  a  radial  line.     The  chord  of 
the  swinging  arc  of  the  roller  center  is  to  be  4  units  and  the  length 
of  the  follower  arm  8  units.     The  follower  arm  to  swing: 

(a)  Out  full  distance  in  90°  on  parabola  curve. 

(b)  In      "          "         "  90°  "  crank 

(c)  And  to  remain  at  rest  for  180°. 

86.  A  different  method  of  construction  from  any  thus  far  em- 
ployed is  used  in  problems  of  this  kind  because  it  gives  the  simplest 
and  most  accurate  layout  for  the  pitch  surface.     Briefly,  the  method 
to  be  used  consists  in  revolving  the  follower  through  the  360°  around 
the  cam  while  the  cam  remains  stationary,  and  drawing  the  follower 
in  a  number  of  its  phases  while  on  the  way  around.     One  of  the 
phases  is  represented  in  full  by  the  dash  lines  do  Y2  Ys  in  Fig.  45. 

87.  The  angle  which  causes  pressure  against  the  follower  bear- 
ings is  also  different  in  this  form  of  cam  from  any  of  the  others.     An 
inspection  of  Fig.  45  will  show  that,  in  general,  the  normal  line  of 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  51 

pressure,  AV  at  A,  between  the  cam  surface  and  the  roller  is  not  at 
right  angles  to  the  position  of  the  follower  arm,  and,  therefore, 
that  the  resultant  total  pressure  has  a  component  along  the  arm, 
tending  to  place  it  in  compression  and  throwing  a  corresponding 
pressure  on  the  follower  bearing  at  C.  The  pressure  angle  at  A  is 
shown  by  —  a,  the  minus  sign  indicating  compression  in  the  swinging 
arm.  When  K\  is  at  K  the  pressure  angle  will  be  -f-c,  the  plus  sign 
indicating  tension  in  the  follower  arm.  A  disadvantage  of  the  sign 
changing  from  +  to  — ,  etc.,  is  that  as  soon  as  the  bearings  wear 
there  will  be  noise  at  that  point. 

88.  The  detail  for  the  construction  of   problem  8  is  taken  up 
by  computing  the  diameter  of  the  pitch  circle  first,  as  in  previous 
problems.     This  computation,  however,  serves  only  as  a  guide,  for 
the  assigned  pressure  angle  will  be  both  increased  and  decreased  by 
amounts  depending  on  the  radius  of  the  follower  arm  and  the  char- 
acteristics of  the  base  curve  which  is  used.     For  computing  the 
pitch  circle  then,  an  assigned  pressure  angle  factor  for  30°  will  be 
assumed  in  the  expectation  that  the  final  maximum  angle  will  not 

4  X  3.46 
exceed  40°.     From  formula   1,   paragraph  29,   d  =  114.6  — ^Tr1- 

4  X  2  72 

=  17.62  for  the  parabola  curve;    and  d  =  114.6  -  -  =  13.86 

yu 

for  the  crank  curve  assignment.  The  radius  of  the  pitch  circle  is 
thus  found  to  be  8.81  units. 

89.  Having  determined  the  radius  0  D,  Fig.  45,  for  the  pitch  circle, 
the  given  chord  of  4  units  is  laid  off  with  equal  parts  on  each  side 
of  D,  thus  locating  the  ends  of  the  swinging  arc  A  V  on  the  radial 
line  0  D  as  required.     With  A  and  V  as  centers  and  a  radius  of 
8  units  for  the  length  of  the  follower  arm,  strike  arcs  which  will 
intersect  at  C  and  give  the  fixed  center  for  the  follower  arm.     The 
arc  A  V,  showing  the  path  of  the  center  of  the  follower  roller  is  now 
drawn. 

90.  Points  on  the  pitch  surface  A  ViA2F  are  found,  in  brief, 
by  revolving  the  arm  C  A  around  0,  swinging  it  a  proper  amount 
on  its  center  C  as  it  revolves.     In  detail  this  is  accomplished  by  lay- 
ing off  the  arc  C  Ce  equal  to  90°,  and  dividing  it  into  a  number  of 
equal  parts,  say  six.     Divide  the  arc  A  J  into  three  unequal  parts, 
as  at  H  and  /,  for  the  parabola  curve.     Lay  off  the  points  L  and  K 
in  the  same  way.     Then  with  C  A  as  a  radius  and  with  Ci,  C2  .  .  . 
as  centers  draw  the  arcs  passing  through  HI,  Ii  .  .  .  .  Again,  with 


52  ELEMENTARY   CAMS 

0  as  a  center,  swing  arcs  through  H,  I  .  .  .  until  they  meet  the 
arcs  already  constructed.  The  intersections  of  these  arcs,  as  at 
Hi,  /i,  Ji  .  .  .  will  be  the  points  on  the  desired  pitch  surface  AVi. 
The  determination  of  the  pitch  surface  for  the  crank  curve  is  found 
by  laying  off  the  second  90°  assignment  from  Ce  to  Ci2  and  dividing 
it  into  six  parts.  The  arc  AI  Vi  is  divided  by  projecting  the  points 
U  W  .  .  .  of  the  crank  circle  to  the  points  Ui  W\  on  the  arc.  The 
constructions  for  the  points  U2,  TF2  .  .  .  are  the  same  as  for  the 
previous  part  of  the  pitch  surface,  as  described  above. 

91.  EXERCISE  PROBLEM  8a.     Required  a  radial  periphery  cam 
to  operate  a  roller  follower  where  the  follower  arm  swings  about  a 
pivot  and  the  two  extremities  of  the  swinging  arc  lie  on  a  radial 
line.     The  30°  pressure  angle  factor  to  be  used  in  computing  the 
pitch  circle  radius.     The  chord  of  the  swinging  arc  to  be  3  units, 
the  arm  9  units  long,  and  to: 

(a)  Swing  out  in  J/£  revolution  on  the  crank  curve  base. 

(b)  Remain  at  rest  for  %  revolution. 

(c)  Swing  in  in  J/£  revolution  on  the  parabola  base. 

92.  PROBLEM  9.     CAM  WITH  SWINGING  FOLLOWER  ARM,  ROLLER 
CONTACT — SWINGING  ARC,  CONTINUED,  PASSES  THROUGH  CENTER  OF 
CAM.     Required  a  radial  periphery  cam  to  operate  a  roller  follower 
where  the  follower  arm  swings  about  a  pivot,  and  where  the  center 
of  the  follower  roller  moves  on  an  arc  which,   continued,  passes 
through  the  center  of  the  cam.     The  chord  of  the  swinging  arc  of 
the  roller  center  is  4  units  and  the  length  of  the  follower  arm  10 
units.     The  follower  arm  to  swing: 

(a)  Out  full  distance  in  90°  on  parabola  curve. 

(b)  In      "         "         "  90°  "  crank 

(c)  To  remain  at  rest  for  180°. 

93.  The  procedure  for  this  problem  is  the  same  as  for  Problem  8 
in  all  respects  except  the  layout  of  the  arc  of  swing  for  the  center 
of  the  follower  roller.     The  pitch  circle  is  drawn  with  radius  0  J, 
Fig.  46. 

With  the  center  of  the  cam  0  and  the  pitch  point  /  as  centers 
draw  arcs  which  intersect  at  C,  the  radius  being  equal  to  the  length 
of  the  follower  arm.  Lay  off  J  A  and  J  V  equal  to  each  other  and 
so  that  a  chord  drawn  from  A  to  V  equals  the  four  units  assigned. 
A  bent  rocker,  A  C  E,  is  introduced  in  Fig.  46  simply  to  change  the 
direction  of  motion. 

94.  EXERCISE  PROBLEM  9a.     Required  a  radial  periphery  cam  to 
operate  a  roller  follower  where  the  follower  arm  swings  about  a  pivot, 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


53 


and  where  the  center  of  the  follower  roller  moves  on  an  arc  which, 
continued,  passes  through  the  center  of  rotation  of  the  cam.  Take 
the  length  of  follower  arm  as  12  units  and  its  angle  of  swing  30°. 
Required  that  the  follower  arm: 

(a)  Swing  out  full  distance  in  ^g  revolution,  on  crank  curve. 

(b)  Remain  at  rest  }^  revolution. 

(c)  Swing  in  full  distance  in  %  revolution,  on  crank  base. 


9>} 


FIG.  45. — PROBLEM  8,  CAM  WITH  SWINGING  FOLLOWER  ARM,  ROLLER  CONTACT — EXTREMI- 
TIES OF  SWINGING  ARC  ON  RADIAL  LINE 

95.  EFFECT  OF  LOCATION  OF  SWINGING  FOLLOWER  ARM  RELATIVELY 
TO  THE  CAM.  When  the  swinging  follower  arm  is  mounted  so  that 
the  extremities  of  the  arc  of  travel  of  roller  center  are  on  a  radial 
line,  as  in  Problem  8,  the  pressure  angles  on  the  out  and  in  strokes 
will  be  approximately  the  same.  When  the  follower  roller  center 


54 


ELEMENTARY   CAMS 


moves  on  an  arc  which,  continued,  passes  through  the  center  of 
the  cam,  as  in  Problem  9,  the  pressure  angle  will  be  larger,  on  the 
average,  on  the  one  stroke  than  on  the  other.  Consequently,  the 
type  shown  in  Problem  8  would  have  an  advantage  where  equal 
amounts  of  work  were  to  be  done  on  both  strokes,  and  the  type 


FIG.  46. — PROBLEM  9,  CAM  WITH  SWINGING  FOLLOWER  ARM,  ROLLER  CONTACT — SWINGING 
ARC,  CONTINUED,  PASSES  THROUGH  CENTER  OF  CAM 

shown  in  Problem  9  would  be  of  advantage  where  heavy  work  was 
to  be  done  on  one  stroke  only.  Either  the  out  or  in  stroke  may 
be  selected  for  heavy  work,  according  to  the  position  taken  for  the 
center  C  or  G  of  the  swinging  arm,  Fig.  46,  the  direction  of  turning 
of  the  cam  being  the  same.  In  many  cases  the  type  shown  in 
Problem  9  allows  the  pressure  angle  to  be  maintained  on  one  of  the 
strokes  so  that  there  is  pressure  in  only  one  direction  on  the  shaft  C. 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  55 

Cams  operate  smoother  when  "running  off"  than  when  "running 
on."  A  cam  is  said  to  be  "running  off"  when  the  point  of  contact 
on  the  working  surface  of  the  cam  is  moving  away  from  the  fixed 
center  of  the  swinging  follower  arm.  A  cam  of  the  type  illustrated 
in  Problem  8  will  have  an  axis  of  symmetry  where  the  same  data 
are  assigned  for  the  out  and  in  stroke,  whereas  the  cam  illustrated 
in  Problem  9  will  be  quite  unsymmetrical  for  same  data. 

96.  POSITIVE-DRIVE   FACE   CAMS.     The   pitch   surfaces   for  face 
cams  are  laid  out  in  exactly  the  same  manner  as  pitch  surfaces  for 
radial  periphery  cams.     The  only  additional  feature  is  that  a  working 
surface  is  drawn  to  touch  each  side  of  the  roller. 

97.  PROBLEM  10.     FACE  CAM  WITH  SWINGING  FOLLOWER.     Con- 
struct a  face  cam  for  a  swinging  follower  arm,  roller  contact.     Arm 
to  be  12  units  long  and  to  swing  through  30°.     Required  that  the 
arm  shall: 

(a)  Swing  full  out  on  the  combination  curve  while  the  cam  makes 
%  revolution. 

(b)  Swing  full  in  on  the  combination  curve  while  the  cam  makes 
%  revolution. 

98.  In  order  to  compute  the  radius  of  the  pitch  circle  it  is  neces- 
sary to  find  the  travel,  or   the   approxi- 
mate travel,  of  the  center  of  the  follower 

roller.  This  is  graphically  done  by  mak- 
ing a  separate  sketch,  as  in  Fig.  47, 
drawing  the  angle  X  Y  Z  equal  to  30°, 
drawing  the  arc  Z  X  with  a  radius  of  12 
units,  and  measuring  the  chord  Z  X, 

Which  is  found  to  be  6.2  units.      Or,  this       FIG.  47.-To  FIND  CHORD  MEAS- 
URE OF  TRAVEL  OF  POINT  ON 

value   may  be  found  tngonometrically,        SWINGING  ARM 

without  any  drawing,  by  taking  12  X  2 

sin   15°  =  6.2.      The    radius    of    the    pitch    circle  will  then  be: 

6.2  X  2.27  X  y  X  — j4  X  -i-  =  6.0  units. 

99.  To  construct  the  cam,  the  value  just  found  is  laid  off  at  0  J, 
Fig.  48,  and  the  pitch  circle  drawn.     With  the  combination  curve 
a  cam  chart,  a  partial  one  at  least,  must  be  drawn.     To  do  this  with 
least  effort,  select  any  point  «/'  in  line  with  the  pitch  point  J  and 
draw  the  line  J'  V  at  the  given  pressure  angle,  30°  in  this  case, 
until  it  is  6.2  units  long.     With  V  as  a  center,  draw  arc  J'  A'  and 
also  draw  a  tangent  to  it  at  J'  and  produce  it  to  S,'  where  R  S'  equals 


56 


ELEMENTARY   CAMS 


one-half  A'  V.  The  curve  A'  Jf  S'  will  be  one-half  of  the  desired 
base  curve  and  will  be  sufficient  to  proceed  with  the  construction  of 
the  cam.  Divide  the  pitch  line,  0-4,  of  the  chart  into  four  equal 
parts  and  draw  verticals  so  locating  H' ',  /',  Kf.  .  .  .  Project  these 
points  to  H,  I,  K  ...  on  the  arc  of  travel  of  the  center  A  of  the 
roller.  This  construction  will  give  practically  a  uniform  swinging 


FIG.  48. — PROBLEM  10,  FACE  CAM  WITH  SWINGING  FOLLOWER 

velocity  to  the  follower  arm  through  twice  the  angle  measured  by 
the  arc  from  J  to  S.  Theoretically,  the  curve  A'  S'  should  be  con- 
structed on  the  cylindrical  surface  A  S  instead  of  on  its  projected 
plane  surface.  It  is,  however,  unnecessary  to  go  into  the  detail 
of  construction  which  this  would  involve  because  the  difference  in 
results  between  it  and  the  more  direct  process  explained  above 
would  be  too  small  in  practical  cases  to  be  measured  by  the  thickness 
of  the  ordinary  pencil  line. 

With  the  points  H,  I,  K  .  .  .  obtained  as  above,  the  remainder 
of  the  construction  is  the  same  in  detail  as  described  in  connection 
with  Problem  8.  The  reference  letters  are  the  same  in  both  figures. 
The  cam  plate,  in  the  face  of  which  the  groove  for  the  roller  is  cut, 
is  made  circular  in  its  boundary  in  order  to  give  better  balance  and 
appearance. 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  57 

100.  EXERCISE  PROBLEM  lOa.  Required  a  face  cam  for  a  swinging 
follower  arm,  roller  contact.     Arm  to  be  10  units  long.     Center  of 
roller  to  swing  through  an  arc  whose  chord  is  4  units,  and  this  arc, 
when  continued,  to  pass  through  center  of  cam.     The  arm  to: 

(a)  Swing  to  the  right  on  combination  curve  while  cam  turns 
180°. 

(b)  Swing  to  the  left  on  combination  curve  while  cam  turns  180°. 

101.  PROBLEM  11.     CAM  WITH  SWINGING  FOLLOWER  ARM,  SLIDING 
SURFACE  CONTACT.     Required  a  radial  periphery  cam  to  operate  a 
swinging  arm  having  a  construction  radius  of  9  units.     Sliding  sur- 
face contact  between  cam  and  follower.     The  arm  to: 

(a)  Swing  up  4  units  on  the  crank  curve  base  while  the  cam 
turns  120°. 

(b)  Swing  down  4  units  on  the  crank  curve  base  while  the  cam 
turns  120°. 

(c)  Remain  at  rest  for  120°. 

102.  This  type  of  cam  and  follower  is  illustrated  in  Fig.  49.     The 
line  of  pressure  between  cam  and  follower  is  always  normal  to  the 
follower  surface  and  consequently  there  is  no  component  of  pressure 
in  the  bearing  at  C  due  to  pressure  angle.     This  cam  is,  therefore, 
independent  of  a  pitch  circle  based  on  pressure  angle,  and  the  pitch 
circle  may  be  taken  any  size.     Where  one  has  no  special  guide  in 
assuming  a  starting  size  for  the  cam,  the  usual  computation  for 
pitch  circle  for  a  30°  pressure  angle  may  give  good  average  results. 
According  to  this,  the  pitch  radius  0  D  will  be, 

4  X  2.72  X  3  X  3^4  X  -^  =  5.2  units 

AV  equals  4  units  and  A  C  equals  9  units.  The  point  A  is 
taken,  for  construction  purposes,  as  a  point  on  the  follower  arm 
where  the  angular  velocity  of  the  arm  is  measured.  It  will  be 
at  the  points  H,  I,  J  ...  on.  the  arc  A  V  at  the  end  of  equal  suc- 
ceeding intervals  of  time. 

103.  The  method  of  constructing  the  cam  in  this  problem  is 
identical    with   the   method   used   in    Problem  8  in  so   far  as  the 
follower  arm  is  swung  around  the  cam,  and  its  position  with  respect 
to  the  cam  center  at  equal  time  intervals  is  drawn.     The  departure 
from  the  method  of  Problem  8  consists  in  drawing  the  cam  outline 
as  an  envelope  to  these  follower-arm  positions.     For  example,  in 
Fig.  49,  at  the  end  of  the  third  time  interval  the  pivot  C  has  been 
revolved  to  €3  and  the  point  A  of  the  follower  arm  has  moved  out 


58  ELEMENTARY   CAMS 

to  Ji.  The  point  Ji  is  found  at  the  intersection  of  two  arcs,  one 
obtained  with  C  A  as  a  radius  and  C3  as  a  center,  and  the  other 
with  0  J  as  a  radius  and  0  as  a  center. 

When  a  number  of  positions  of  the  follower  arm,  such  as  €3  J\t 
have  been  obtained,  the  smoothest  possible  curve  is  drawn  tangent 
successively  to  each  of  them,  and  this  curve  is  the  working  surface 


c. 
FIG.  49.— PROBLEM  ll,  CAM  WITH  SWINGING  FOLLOWER  ARM,  SLIDING  CONTACT 

of  the  cam.  This  curve  is  tangent  to  C&  J\  at  M ,  and  if  the  distance 
Cs  M  is  laid  off  at  C  MI,  the  point  MI  will  be  the  actual  point  of 
tangency  between  the  cam  surface  and  follower  arm  when  the  arm 
is  halfway  through  its  swing,  or  when  A  is  at  J.  Similarly  when 
Cg  is  at  C  the  point  of  tangency  between  cam  and  follower  arm  will 
be  at  Ni. 

104.  The  locus  of  the  point  of  contact  between  the  cam  and 
follower,  relatively  to  the  frame  of  the  machine,  is  shown  by  the 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  59 

dash  closed  curve  through  MI  and  NI.  By  drawing  arcs  tangent 
at  the  extremities  of  this  dash  curve,  using  C  as  a  center  in  both 
cases,  the  points  F  and  G  on  the  follower  surface  are  obtained  and 
the  distance  F  G  will  be  the  part  of  the  follower  exposed  to  wear 
from  the  rubbing  of  the  cam.  This  part  of  the  follower  arm  may  be 
designed  with  a  shoe,  as  indicated,  which  may  be  replaced  when 
worn. 

105.  It  should  be  specially  noted  that  the  shortest  radius  of  the 
cam  is  not  0  A,  but  0  B.     The  point  B  is  found  by  drawing  a  per- 
pendicular to  C  G  through  0. 

The  very  decided  lack  of  symmetry  should  also  be  noted,  the 
curve  B  L  being  used  to  lift  the  arm,  and  the  curve  L  E  to  lower 
the  arm,  the  swinging  velocities  of  the  arm  being  the  same  in  both 
directions. 

106.  DATA  LIMITED  FOR  FOLLOWERS  WITH  SLIDING  SURFACE  CON- 
TACT.    The  data  for  this  type  of  cam  construction  are  extremely 
limited  when  the  swinging  velocity  of  the  arm  is  assigned.     The 
limitations  are  that  the  working  surface  of  the  cam  must  be  drawn 
tangent  to  every  construction  line  in  succession,  and  that  it  must  be 
convex  externally  at  all  points.     In  most  arbitrary  assignments  of 
data  the  construction  line  through  Cg,  for  example,  would  intersect 
the  line  through  C?  before  it  cut  the  line  through  C&.     In  this  case 
it  would  be  impossible  to  draw  a  smooth  working  curve  tangent, 
successively,  to  the  lines  through  C?,  Cg,  and  Cg.     This  is  illustrated 
more  clearly  in  Fig.  51  and  will  be  more  evident  after  the  limiting 
case  is  described. 

The  limiting  case  for  flat  surface  followers  with  sliding  contact 
occurs  where  three  or  more  of  the  construction  lines  meet  in  a  point, 
as  at  N  in  Fig.  50.  In  this  case  the  working  surface  of  the  cam 


FIG.  50. — LIMITING  CASE  FOR  STRAIGHT  EDGE     FIG.  51. — IMPOSSIBLE  CASE  TOR  STRAIGHT 
FOLLOWER  WITH  SLIDING  CONTACT  EDGE  FOLLOWER  WITH  SLIDING  CONTACT 


60  ELEMENTARY   CAMS 

would  have  a  sharp  edge.  In  this  type  of  cam  it  is  necessary  to 
use  more  construction  lines  than  in  other  types,  because  it  is  pos- 
sible to  have  the  construction  lines  so  far  apart  that  such  a  case  as 
is  shown  in  Fig.  51  might  not  evidence  itself  at  all.  For  example, 
if  the  distance  Cg  €7  were  the  unit  space  for  construction  lines,  in- 
stead of  Cg  Cg,  the  smooth  convex  curve  F  N  L  could  be  drawn 
tangent  to  lines  through  Cg,  €7  .  .  .  without  the  error  showing 
itself. 

107.  If  it  is  required  of  this  cam  only  that  it  shall  swing  a  follower 
arm  through  a  given  angle  in  a  given  time,  without  regard  to  the 


FIG.  50. — (Duplicate.)     LIMITING  CASE  FOR         FIG.  51. — (Duplicate.)    IMPOSSIBLE  CASE 
STRAIGHT  EDGE  FOLLOWER  WITH  SLIDING  FOR  STRAIGHT  EDGE  FOLLOWER  WITH 

CONTACT  SLIDING  CONTACT 

intermediate  velocities  of  the  arm,  it  may  be  as  widely  used  as  any 
other  type  of  cam.  In  this  case  only  the  innermost  and  outermost 
positions  of  the  arm  would  be  drawn,  as  at  C  A,  C&  Vi,  and  C&E, 
Fig.  49,  and  a  smooth  convex  curve  drawn  tangent  to  these  lines. 
Such  construction,  however,  might  give  an  irregular  or  jerky  motion 
to  the  follower.  Whether  it  did  or  not  could  be  readily  determined 
by  laying  off  a  number  of  equal  divisions,  as  at  Ci,  C2  .  .  .  Ci2; 
drawing  lines,  such  as  €3  Ji,  tangent  to  the  assumed  smooth  convex 
working  surface;  and  revolving  C3  J\  back  to  C  J.  After  doing 
this  with  other  construction  lines  a  series  of  points,  such  as  H,  I,  J 
.  .  .  would  be  determined  and  the  spaces  between  them  would 
represent  the  distances  traveled  by  A  on  the  follower  arm  during 
successive  equal  intervals  of  time. 

108.  EXERCISE  PROBLEM  lla.  Required  a  radial  periphery  cam 
for  a  swinging  follower  arm,  sliding  surface  contact.  Arm  to  be 
10  units  long  to  the  point  which  is  used  to  measure  the  angular 
velocity,  and  this  point  to  move  through  an  arc  which  is  measured 
by  a  chord  of  4  units.  The  arm  is  to: 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  61 

(a)  Swing  full  out  with  uniform  acceleration  and  retardation 
while  the  cam  turns  Y%  revolution. 

(b)  Swing  in  with  the  same  angular  motion  in  ^  revolution. 

(c)  Remain  stationary  for  %  revolution  of  the  cam. 

109.  TOE  AND  WIPER  CAMS.     In  this  form  of  cam  construction 
the  cam  or  "wiper"  0  C,  Fig.  52,  oscillates  or  swings  back  and 
forth  through  an  angle  of  120°  or  less,  instead  of  rotating  con- 
tinuously the  full  360°  as  it  does  in  all  cams  thus  far  considered. 
The  follower  or  "toe"  A  W  is  usually  a  narrow  flat  strip  resting  on 
the  curved  periphery  of  the  cam,  and  moving  straight  up  and  down. 
There  is  sliding  action  between  the  wiper  and  the  toe. 

110.  PROBLEM  12.     TOE  AND  WIPER  CAM.     Required  a  wiper  cam 
to  operate  a  flat  toe  follower  which  shall  move: 

(a)  Up  4  units  with  uniform  acceleration  all  the  way  while  the 
cam  turns  counterclockwise  45°  with  uniform  angular  velocity. 

(b)  Down  4  units  with  uniform  retardation  all  the  way  while 
the  cam  turns  clockwise  45°  with  uniform  angular  velocity. 

111.  The  detail  of  construction  for  this  class  of  problem  is  iden- 
tical with  that    described   for   the   mushroom    cam  in  Problem  7, 
it  being  observed  that  the  two  cams  differ  only  in  that  the  mush- 
room cam  turns  through  the  full  360°  instead  of  45°  as  in  this  problem, 
and  the  mushroom  follower  is  circular  instead  of  rectangular.     Neither 
of  these  differences  nor  the  offset  of  the  mushroom  follower  affect 
the  similarity  of  construction  for  the  two  types  of  cams.     There- 
fore, only  a  brief  review  of  the  general  method  of  construction  for 
the  present  problem  will  be  given  here. 

112.  Inasmuch  as  the  line  of  pressure  between  cam  and  follower 
is  always  parallel  to  the  direction  of  motion  of  the  follower  in  prob- 
lems such  as  this,  there  is  no  pressure  angle  in  the  ordinary  sense. 
If  a  computation  for  size  of  cam  is  made  in  the  usual  way,  the  radius 
of  the  pitch  circle  will  figure  to  be  unnecessarily  large,  due  princi- 
pally to  the  fact  that  only  a  45°  degree  turn  of  the  cam  is  allowed 
for  the  upward  motion  of  the  follower. 

A  radius  0  A,  Fig.  52,  which  allows  for  radius  of  shaft,  thickness 
of  hub,  etc.,  is  assumed,  and  the  follower  motion  of  4  units  is  laid 
off  at  A  V.  This  distance  is  divided  into  four  unequal  parts  at 
H,  I  .  .  .  which  are  to  each  other  as  1,  3,  5,  and  7,  thus  giving 
uniform  acceleration  all  the  way  up.  The  angle  A  0  B  of  the  cam 
is  laid  off  45°  and  is  divided  into  four  equal  time  parts.  The  follower 
or  toe  surface  A  W  is  then  moved  up  the  distance  A  H  and  revolved 
through  the  angle  A  0 1  to  the  position  HI  H2  which  is  marked.  Simi- 


62 


ELEMENTARY   CAMS 


larly  A  W  is  next  moved  to  7  73  and  revolved  to  7i  72.  The  smooth- 
est possible  convex  curve  is  then  drawn  to  the  lines  HI  H2)  Ii  72  .  .  . 
and  this  curve  becomes  the  working  surface  of  the  wiper. 

The  necessary  working  length  for  the  wiper  is  found  to  be  A  Vz, 
and,  adding  a  small  arbitrary  distance,  V2  C,  the  total  length  is  taken 


FIG.  52.— PROBLEM  12,  TOE  AND  WIPER  CAM 

as  A  C.  The  total  length  of  the  toe  A  W  will  be  equal  to  Vi  C. 
The  long  dash  lines  in  Fig.  52  indicate  the  highest  position  of  the 
toe  and  wiper,  and  the  short  dash-line  curve  marks  the  locus  of 
contact  between  the  wiper  and  toe.  This  curve  is  obtained  by 
making,  for  example,  /  J3  equal  to  Ji  J2. 

113.  MODIFICATIONS  OF  THE  TOE  AND  WIPER  CAM.  The  toe  and 
wiper  cam  constructions  are  commonly  used.  In  the  present  ele- 
mentary problems  the  cam  or  wiper  is  assumed  to  oscillate  with 
uniform  angular  velocity,  whereas  in  practice  it  usually  has  a  variable 
angular  velocity  due  to  the  fact  that  it  is  operated  through  a  rod 
which  is  connected  at  the  driving  end  to  a  crank  pin  or  eccentric 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  63 

whose  diameter  of  action  corresponds  to  the  swing  of  the  wiper 
cam.  The  follower  toe  may  be  built  with  a  curved  instead  of  a 
straight  line,  by  a  slight  modification  in  detail  which  consists  in  draw- 
ing the  curved  toe  line  in  place  of  the  straight  lines,  HI  H2,  /i  /2  .  .  . 
as  shown  in  Fig.  52.  These  points,  together  with  a  consideration  of 
the  amount  of  slip  between  the  surfaces  in  this  type  of  cam  and  a 
discussion  of  the  necessary  modification  to  secure  pure  rolling  in 
cams  of  this  general  appearance,  are  subjects  for  more  advanced 
work  than  is  covered  by  the  present  elementary  problems. 

114.  EXERCISE  PROBLEM  12a.     Required  a  wiper  cam  to  operate 
a  flat  toe  follower  which  shall  move: 

(a)  Up  3  units  with  uniform  velocity  while  the  cam  turns  60° 
in  a  counterclockwise  direction  with  uniform  angular  velocity. 

(b)  Down  3  units  with  uniform  velocity  while  the  cam  turns  60° 
in  a  clockwise  direction  with  uniform  angular  velocity. 

115.  YOKE  CAMS.     Yoke  cams  are  simple  radial  periphery  cams 
in  which  two  points  of  the  follower,  instead  of  one,  are  in  contact 
with  the  working  surface.     The  contact  points  are  usually  diametri- 
cally opposite  to  each  other.     Roller  contact  is  generally  used  and 
the  centers  of  the  rollers  are  a  fixed  distance  apart.     The  yoke  cam 
gives  positive  motion  in  both  directions,  and  does  not  depend  on  a 
spring  or  on  gravity  to  return  the  follower  as  do  all  other  cams 
thus  far  considered,  excepting  the  face  cam. 

116.  PROBLEM  13.     SINGLE-DISK  YOKE  CAM.     Required  a  single 
radial  cam  to  operate  a  yoke  follower  with  a  maximum  pressure 
angle  of  30°: 

(a)  Out  4  units  in  45°  turn  of  the  cam,  on  crank  curve. 

(b)  In    4     "      "  90°     "     "    "       "      "       "         " 

(c)  Out  4     "      "  45°     "     "    "       "      "       "         " 

117.  With  a  single  radial  cam  for  a  yoke  follower,  data  may  be 
assigned  only  within  the  first  180°.  The  reason  for  this  will  appear 
presently. 

Compute  the  radius  of  pitch  circle  as  in  ordinary  radial  cam 
problems.  It  is  found  to  be  13.86  units  and  is  laid  off  at  0  D,  Fig.  53. 
The  pitch  surface,  A  DI  V\  AI  V2,  is  found  in  the  usual  way.  Then 
the  diametral  distance,  AV2,  will  be  the  fixed  distance  between  the 
centers  of  the  rollers,  and  if  this  distance  is  laid  off  on  diametral  lines, 
as  from  /i,  KI  .  .  .,  the  points  W,  X  ...  on  the  complementary 
pitch  surface  will  be  located.  A  size  of  roller  A  B  is  next  assumed 
and  the  working  surface  B  Bz  is  constructed.  The  maximum  radius 
of  the  working  surface  is  finally  located,  as  at  0  B%.  A  small  amount 


64 


ELEMENTARY   CAMS 


is  added  to  this  for  clearance  and  the  total  laid  off  at  0  Z,  thus  giving 
the  width  of  yoke  necessary  for  an  enclosed  cam. 

118.  LIMITED  APPLICATION  OF  SINGLE-DISK  YOKE  CAM.  In  yoke 
cams  constructed  from  a  single  disk  the  data  are  limited  in  two  ways : 

First,  data  can  be  assigned  for  the  first  180°  only,  because  the 
pitch  surface  for  the  second  180°  must  be  complementary  to  the 
pitch  surface  in  the  first  180°. 

Secondly,  the  complementary  pitch  surface  cannot  approach  any 
nearer  to  the  center  of  rotation  of  the  cam  than  does  the  pitch  surface 


— 


"-rpr 


±mi 


R' 


FIG.  53.— PROBLEM  13,  SINGLE-DISK  YOKE  CAM 

in  the  first  180°,  otherwise  the  follower  will  have  a  greater  motion 
than  that  which  was  assigned  to  it. 

To  illustrate  this  second  case,  assume  that  item  (c)  had  been 
changed  in  the  data  for  Problem  13  so  as  to  specify  that  the  follower 
should  remain  at  rest  while  the  cam  turns  45°.  Then  the  pitch 
surface  of  the  cam  for  the  first  180°  would  have  been  AViAiC, 
Fig.  54,  instead  of  AVi  At  V2.  The  diametral  distance  A  C  would 
then  have  been  the  distance  between  roller  centers,  and  would  have 
been  also  the  distance  used  in  determining  the  complementary 
pitch  surface  C  EI  A3  A  which,  it  will  be  noted,  approaches  closer 
to  0  than  does  AViC.  When  EI  of  the  complementary  surface 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS  65 

reaches  the  center  line  0  D,  the  center  A  of  the  roller  will  be  at  E 
and  the  roller  will  have  traveled  the  distance  A  E  in  addition  to  the 
travel  A  V  which  was  assigned.  Furthermore,  the  pressure  angle 
will  be  very  high  when  F  crosses  the  line  0  D2.  With  the  data 
which  gives  the  pitch  surface  A  Vi  C,  the  yoke  follower  will  move 
just  twice  the  assigned  distance.  This  double  motion  will  not  be 
continuous,  as  the  follower 
will  be  at  rest  for  a  definite 
period  represented  by  AiC. 
Even  if  the  data  were  such 
that  AI  should  fall  at  C  there 
would  be  a  momentary  pe- 
riod of  rest  for  the  follower 
at  the  middle  of  its  stroke. 
Summing  up,  the  desired 
travel,  pressure  angle,  and 
follower  velocity  will  be  ob- 
tained in  single-disk  yoke 
cams,  only  when  the  data 
are  such  as  to  have  the  fol-  _ 

PIG.  54.— ILLUSTRATING  LIMITED  APPLICATION  OF 
lower  at    the    extreme    Oppo-  SINGLE-DISK  YOKE  CAM 

site  ends  of  its  stroke  at  the 

zero  and  180°  phases.     In  other  cases  increased  travel,  increased 

pressure   angle,  and  irregular  follower  velocities  will   have  to   be 

considered. 

All  of  the  limitations  of  the  single-disk  yoke  cam  may  be  avoided 
by  using  the  double  disk  cam  as  illustrated  in  Problem  14. 

119.  EXERCISE  PROBLEM  13a.     Required  a  single-disk  radial  cam 
to  operate  a  yoke  follower  with  a  maximum  pressure  angle  of  30°: 

(a)  In     6  units  in  60°  turn  of  the  cam  on  parabola  curve. 

(b)  Out  6     "      "  45°     "     "    "       "     "         "  " 

(c)  At  rest  for        30°     "     "    "       " 

(d)  In     6  units  in  45°     "     "    "       "     " 

120.  PROBLEM  14.     DOUBLE-DISK  YOKE  CAM.    Required  a  double- 
disk  cam  to  operate  a  yoke  follower  with  a  maximum  pressure  angle 
of  30°: 

(a)  To  the  right  6  units  in  150°  turn  of  the  cam,  on  the  crank 
curve. 

(b)  To  the  left  6  units  in  90°  turn  of  the  cam,  on  the  crank  curve. 

(c)  To  remain  stationary  for  120°  turn  of  the  cam. 


66 


ELEMENTARY   CAMS 


121.  The  detail  of  construction  for  the  primary  disk  is  the  same 
as  in  previous  problems  involving  radial  cams.     In  this  problem,  then, 

the  radius  of  the  pitch  circle  is  6  X  2.72  X  4  X  ITTT  X  -^  =  10.4 

O.14         Z 

units  and  this  is  laid  off  at  0  D,  Fig.  55.  The  forward  driving  pitch 
surface,  A  HI  Vi  Ii  A,  is  constructed  in  the  regular  way  as  indicated 
by  the  construction  lines. 

122.  The  diameter  D  C  of  the  pitch  circle  is  next  taken  as  a 
constant  and  its  length  is  laid  off  on  diametral  lines  from  successive 


PIG.  55.— PROBLEM  14,  DOUBLE-DISK  YOKE  CAMS,  DETAIL  CONSTRUCTION 


points  on  the  primary  pitch  surface,  thus  giving  the  secondary  or 
return  pitch  surface.  For  example,  the  point  P  on  the  secondary 
surface  is  found  by  making  A  P  =  D  C;  the  point  M  by  making 
HI  M  =  D  C.  .  .  .  This  second  cam  disk  has  a  pressure  angle  of 
30°  at  Z>4,  the  same  as  the  primary  disk  has  at  D2.  Had  any  diam- 
etral length  other  than  D  C  been  taken  in  this  problem  as  a  constant 
for  constructing  the  second  cam,  the  pressure  angle  at  D4  would 
have  been  greater  or  less  than  the  assigned  30°.  It  does  not  follow 
that  the  diameter  of  the  pitch  circle  should  be  used  as  a  constant 
for  generating  the  complementary  cam.  The  determining  factor, 
in  selecting  a  constant  diametral  length  is  that  the  maximum  pres- 
sure angle  on  the  second  cam  should  not  exceed  the  assigned  value. 


CAM  PROBLEMS  AND  EXERCISE  PROBLEMS 


67 


123.  To  avoid  intricate  line  work,  only  the  detail  drawing  for 
the  construction  of  the  pitch  surfaces  for  this  problem  is  shown 
in  Fig.  55.  The  pitch  surfaces  are  then  redrawn  in  Fig.  56  and 
the  working  surfaces  and  the  yoke  constructed. 

The  working  surface  of  the  primary  or  forward-driving  cam  is 
shown  at  B  E  F  G  B,  Fig.  56,  and  is  constructed  in  the  same  way  as 


r 

-) 

\w          \ 

|         \T' 

A 

r 

i  i 

\s> 

! 

! 

I 

j 

"h 

[ 

j 

l.,,| 

H  >             Z* 

i 

i 

\B>      | 

3  r^ 

FIG.  56. — PROBLEM  14,  DOUBLE-DISK  YOKE  CAMS  SHOWING  STRAP  YOKE  AND  ROLLERS 

in  previous  problems  by  drawing  it  as  an  envelope  to  successive 
roller  positions.  The  working  surface  of  the  return  cam  is  shown 
at  S  Q  R  S.  A  special  caution  to  be  observed  at  this  point  is  that 
the  working  surface  of  the  second  cam  cannot  be  obtained  directly 
from  the  working  surface  of  the  first  cam  by  using  the  diametral 
constant;  the  second  cam  pitch  surface  must  be  obtained  first. 

124.  The  form  of  yoke  in  yoke  cams  may  vary,  as  illustrated  for 
example  by  the  box  type  which  encloses  the  cam,  Fig.  53,  and  by 
the  strap  type,  Fig.  56.  In  the  latter  illustration  the  strap  W  X 
has  a  longitudinal  slot  T  U  permitting  it  to  move  back  and  forth 
astride  the  shaft  without  interference.  The  guide  arms  of  the 


68  ELEMENTARY   CAMS 

yoke  are  shown  at  Y  and  Z.  In  all  yoke  constructions  it  is  desirable 
to  have  all  the  forces  acting  in  as  nearly  a  straight  line,  or  in  a  plane, 
as  possible.  In  Fig.  53  this  is  obtained,  as  may  be  noted  in  the 
top  view  where  the  longitudinal  center  lines  of  cam  disk,  cam  roller, 
yoke  and  yoke  guides  are  all  in  the  same  plane.  In  Fig.  56  the  yoke 
guides,  Yr  and  Z',  are  placed  in  a  line  lying  between  the  cam  disks, 
Bf  and  $',  so  as  to  have  the  forces  balanced  to  a  greater  degree  than 
they  would  be  if  the  guides  were  in  line  with  the  strap  Wf  Xf. 

125.  EXERCISE  PROBLEM  14a.  Required  a  double-disk  cam  to 
operate  a  yoke  follower  with  a  maximum  pressure  angle  of  30°,  as 
follows : 

(a)  To  the  right  4  units  in    90°  on  the  parabola  base. 

(b)  Dwell  for  30°. 

(c)  To  the  right  4     "      "  105°  "     " 

(d)  "    "   left    8     "      "  135°  "     " 

126  PROBLEM  15.  CYLINDRICAL  CAM  WITH  FOLLOWER  THAT 
MOVES  IN  A  STRAIGHT  LINE.  Required  a  cylindrical  cam  to  operate 
a  reciprocating  follower  rod: 

(a)  To  the  right  4  units  in  120°  on  the  crank  curve. 

(b)  "     "    left    4     "      "  120°   "     " 

(c)  "  dwell  120°. 

The  maximum  surface  pressure  angle  to  be  30°. 

127.  The  size  of  cylinder  is  found  by  a  computation  similar  to 
that  for  radial  cams,  and  in  this  problem  the  radius  of  the  cylinder  is, 

4  X  2.72  X  3  X  O-TJ  X  o-  =  5.2  units. 

This  distance  is  laid  off  at  Of  Ar  in  Fig.  57,  and  the  circle  drawn. 
The  distance  A  V,  the  travel  of  the  follower,  is  laid  off  equal  to 
4  units  and  subdivided,  according  to  the  crank  circle,  at  H,  I  .  .  . 
The  radius  of  the  follower  pin  is  assumed  as  at  A  S  and  this  distance 
is  laid  off  at  S  C,  thus  locating  the  edge  of  the  cylinder.  Make 
V  D  equal  to  A  C.  The  circle  representing  the  cylinder  is  next 
divided  into  three  120°  divisions  at  Af,  Mf ,  and  Qf,  as  specified. 
Af  Mr  is  divided  into  six  equal  parts  by  the  points  Hf,  I'  .  .  . 
which  are  projected  over  to  meet  the  vertical  construction  lines 
through  H,  I  .  .  .  at  H^  72.  .  .  .  The  latter  points  mark  a  curve  on 
the  surface  of  the  cylinder.  This  curve  is  a  guide  for  the  center  of 
the  tool  which  cuts  the  groove.  The  finding  of  this  curve  and  the 
construction  of  the  follower  pin  and  rod  constitute  the  remaining 
essential  work  on  this  problem.  If  it  is  desired  to  show  the  groove 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


69 


itself,  the  directions  in  paragraph  134  will  give  an  approximate 
method.  The  follower  pin  is  attached  to  a  follower  rod  X  which 
is  guided  by  the  bearings  F  and  Z.  The  assigned  pressure  angle 
of  30°  is  shown  in  its  true  size  at  D  J  G]  J  D  being  parallel  to  the 
direction  of  motion  of  the  follower  rod,  and  J  G  being  a  normal 
to  the  cutting-tool  curve  M  N  J  P.  .  .  .  In  general,  the  pressure 
angle  will  not  show  in  its  true  size,  and  if  it  is  then  desired  to  illus- 
trate it,  the  cylinder  may,  in  effect,  be  revolved  until  the  correct 
point  of  the  cutting-tool  curve  is  projected  on  the  horizontal  center 
line.  The  exact  point  E  where  the  cutting-tool  curve  comes  tangent 
to  the  bottom  line  of  the  cylinder  may  be  found  by  locating  Ei 
relatively  to  KI  and  LI,  the  same  as  Ef  is  located  relatively  to  Kr 
and  Z/,  and  projecting  EI  down  to  E. 

A  small  clearance  is  allowed  between  the  end  Bf  of  the  pin  and  the 
inner  surface  of  the  groove,  which  is  represented  by  the  dash  circle 
passing  through  Ff . 

128.  REFINEMENTS  IN  CYLINDRICAL  CAM  DESIGN.  It  will  be 
noted  that  the  " maximum  surface  pressure  angle"  was  given  in  the 
data  for  this  problem  instead  of  the  term  "  maximum  pressure 
angle"  that  has  been  used  thus  far.  The  reason  for  this  is  that 
the  pressure  angle  varies  along  the  length  of  the  pin  and  is  always 
greatest  at  the  outer  end,  that  is,  at  the  point  B  in  Fig.  57.  This  is 
not  important  in  most  practical  cases.  Further,  the  term  "  pitch 
cylinder"  is  not  mentioned  in  the  simple  form  of  practical  construc- 


FIG.  57.— PROBLEM  15,  CYLINDRICAL  CAM  WITH  FOLLOWER  SLIDING  IN  A  STRAIGHT  LINE 


70  ELEMENTARY    CAMS 

tion  here  used.  Since  the  pitch  cylinder  should  pass  through  the 
point  where  maximum  pressure  angle  exists,  the  pitch  cylinder 
in  cams  of  this  type  would  be  one  having  a  radius  0'  Bf.  The  pitch 
surface  of  the  cylindrical  cam  would  be  a  warped  surface,  known 
as  the  right  helicoid,  and  the  intersection  of  this  surface  with  the 
surface  of  the  cylinder  is  the  curve  R  A  E  P  R  and  is  the  guide 
curve  for  the  cutting  tool  in  milling  out  the  groove  for  the  pin. 
The  sides  of  the  groove  are  the  working  surfaces  of  the  cam;  they 
are  indicated  in  the  sectioned  part  of  the  front  view  of  Fig.  57. 

More  exact  methods  for  drawing  the  sides  of  the  groove  in  a 
cylindrical  cam,  together  with  a  more  exact  method  for  determining 
the  maximum  pressure  angle,  involve  a  knowledge  of  projections 
and  an  intricacy  in  drawing  that  make  such  work  a  proper  subject 
for  advanced  study,  and  it  will  therefore  be  omitted  in  this  elemen- 
tary treatment,  as  it  is  totally  unnecessary  in  most  practical  work. 

129.  EXERCISE  PROBLEM   15a.     Required  a  cylindrical  cam  to 
operate  a  sliding  follower  rod,  with  a  maximum  pressure  angle  of  40° : 

(a)  6  units  to  the  right  in    90°  on  the  parabola  base. 

(b)  6     "      "     "    left     "  270°   "     " 

130.  PROBLEM  16.     CYLINDRICAL  CAM  WITH  SWINGING  FOLLOWER. 
Required  a  cylindrical  cam  to  operate  a  swinging  follower  arm: 

(a)  To  the  left     40°  in  120°  turn  of  the  cam  on  the  crank  curve. 

(b)  "    "    right  40°  "  120°     "      "     "      "      "     " 

(c)  Dwell  for  120°  turn  of  the  cam. 

The  length  of  the  follower  arm  is  to  be  9  units  and  the  approximate 
maximum  pressure  angle  is  to  be  30°. 

131.  The  diameter  for  the  cylindrical  surface  is  found  in  the 
same  manner  as  the  diameter  of  the  pitch  circle  in  radial  periphery 
cams.     The  data  in  this  problem  do  not  give  directly  the  travel  of 

the  follower  and  so  this  value  must  be 
found  first.  The  chord  of  a  40°  arc  hav- 
ing 9  units  radius  will  be 


9  X  2  X  sin  20°  =  9  X  2  X  .342  =  6.2  units. 
If   a   trigonometrical   table  is  not   at 

FIG.  58.  —  DETERMINING   FOL-       11,1  ,          •, 

LOWER  TRAVEL  IN  SAVING  ING      hand   the   arc   may   be   drawn   out  as  in 


FiS-  58  where  half  the  given  angle  is  laid 
out  at  AYJ  by  the  simple  expedient  of 

subdividing  a  30°  arc  by  means  of  a  dividers.  The  half  chord 
A  D  is  drawn  and  measured.  It  is  equal  to  3.1  units,  thus  mak- 
ing the  chord  of  the  whole  arc  of  travel  equal  to  6.2  units. 


CAM  PROBLEMS  AND  EXERCISE  PROBLEMS 


71 


This  value  is  used  in  obtaining  the  diameter  of  the  surface  of  the  cyl- 
inder as  follows: 

1 


6.2  X  2.72  X  3  X 


3.14 


=  16.12  units. 


The  circle  A1  Q'  M',  Fig.  59,  is  drawn  with  a  radius  of  8.06  units. 

132.  The  120°  angles  assigned  in  the  data  are  next  laid  out  but 
not  from  the  center  line  0  Rf  as  in  previous  problems.  In  mechan- 
isms of  all  kinds  where  there  is  a  swinging  follower,  it  is  a  rule,  unless 
otherwise  specified,  that  the  swinging  pin  should  be  the  same  dis- 


FIG.  59. — PROBLEM  16,  CYLINDRICAL  CAM  WITH  SWINGING  FOLLOWER  ARM 

tance  above  a  center  line  at  the  middle  of  its  swing  as  it  is  below 
at  the  two  extremities  of  its  swing.  In  this  case,  then,  the  point  G, 
Fig.  58,  will  be  marked  midway  between  J  and  D  and  the  distance 
G  J  laid  off  at  G  J  in  Fig.  59.  Y  will  be  the  center  of  swing  of  the 
follower  arm  and  the  arc  of  swing  of  the  follower  pin  will  be  A  J  V. 
J  will  be  as  much  above  the  center  line  as  A  and  V  are  below.  The 
practical  advantage  of  this  detail  in  the  layout  is  that  it  gives  a 
maximum  bearing  length  between  the  follower  pin  and  the  side  of 
the  groove. 

133.  The  arc  A  J  V,  Fig.  59,  is  next  divided  at  the  points  marked 
H,  I  .  .  .  according  to  the  crank  curve  assignment,  and  vertical 
construction  lines  are  drawn  through  these  points. 

The  point  A  is  now  projected  to  A'  and  the  radial  line,  Af  0,  is 
drawn.  This  becomes  the  base  line  from  which  to  lay  off  the  three 


72  ELEMENTARY   CAMS 

assigned  timing  angles  of  120°,  as  shown  at  A'  0  M',  Mf  0  Q',  and 
Qf  0  A'.  The  arc  A'  Mf  is  next  divided  into  the  desired  number 
of  equal  construction  parts,  as  at  Hs,  Is,  Js-  .  .  . 

When  Hz  reaches  A',  the  pin  A  will  have  swung  not  only  over 
to  Hy  but  it  will  have  moved  up  the  distance  A'  H'  measured  on  the 
surface  of  the  cylinder.  Therefore,  when  HZ  reaches  A',  it  is  the 
line  through  H$  (H3  H5  =  A'  H')  on  the  groove  center  line  that  will 
be  in  contact  with  the  pin  center  line.  For  this  reason  #5,  instead 
of  HZ,  is  projected  over  to  meet  the  construction  line  at  H2.  This 
latter  point  is  on  the  guide  curve  for  the  cutting  tool  on  the  surface 
of  the  cylinder.  Other  points  are  found  in  the  same  way.  Time 
may  be  saved  by  marking  the  points  A'  H'  I'  Jf  on  the  straight  edge 
of  a  piece  of  paper  and  transferring  these  marks  at  one  time  so  as 
to  obtain  the  points  75,  J5  .  .  .  P5.  .  .  . 

134.  If  it  is  required  to  show  the  surface  bounding  lines  of  the 
side  of  the  groove  it  may  be  done  quickly,  although  approximately, 
by  laying  off  on  a  horizontal  line,  as  at  /2,  the  points  74  and  I&  at 
distances  equal  to  the  radius  of  the  pin.     These  will  represent  points 
on  the  curve.     If  it  is  required  to  show  the  bottom  lines  of  the 
groove   it   may   be   done   by  projecting  from  77  and  finding,  for 
example,  the  point  Is  in  the  same  way  as  74  was  found. 

135.  EXERCISE  PROBLEM   16a.     Required  a  cylindrical  cam  to 
operate  a  swinging  follower  arm: 

(a)  To  the  right  6  units  (measured  on  chord  of  follower  pin  arc) 
while  cam  turns  150°. 

(b)  Dwell  while  cam  turns  120°. 

(c)  To  the  left  6  units  while  cam  turns  90°. 

The  follower  arm  to  be  8  units  long  and  its  rate  of  swinging  to  be 
controlled  by  the  crank  curve  with  a  maximum  approximate  pres- 
sure angle  of  40°. 

136.  CHART  METHOD  FOR  LAYING  OUT  A  CYLINDRICAL  CAM  WITH 
A  SWINGING  FOLLOWER  ARM.     This  method  is  illustrated  in  Figs.  60 
and'  61.     The  data  in  this  problem  will  be  taken  the  same  as  in 
Problem  16,  namely,  that  a  follower  arm  of  9  units  length  shall: 
Swing  through  an  angle  of  40°  to  the  left  while  the  cam  turns  120°; 
through  the  same  angle  to  the  right  while  the  cam  turns  120°,  on 
the  crank  curve  in  both  directions;    remain  stationary  while  the 
cam  turns  120°.     The  maximum  pressure  angle  is  to  be  approxi- 
mately 30°. 

137.  To  find  the  length  of  the  chart,  the  chord  that  measures 
the  arc  of  swing  of  the  follower  pin  is  first  determined  to  be  6.2 


CAM    PROBLEMS    AND    EXERCISE    PROBLEMS 


73 


1L 


1M. 


units  as  explained  in  paragraph  131.  The  length 
of  chart  is 

6.2  X  2.72  X  3  =  50.6  units, 

and  this  is  laid  off  at  J  Jlt  Fig.  60.  The  length 
of  the  follower  arm  is  then  laid  off  at  J  Y,  and 
the  follower-pin  arc  A  V  drawn.  This  arc  is 
subdivided  at  H,  I  .  .  .  according  to  the  crank 
curve.  The  distance  Y  Y&  is  then  laid  off  to 
represent  120°  and  its  length  will  be  equal  to 
one-third  the  length  of  the  chart.  As  many 
construction  points  as  were  used  from  A  to  V 
are  then  laid  off  between  Y  and  F6.  With 
these  as  centers  and  YA  as  a  radius  draw  a  series 
of  arcs  to  which  the  points  H,  I  .  .  .  are  pro- 
jected, thus  giving  the  base  curve  through  the 
points  HI,  /i.  .  .  .  Tangent  to  the  series  of 
arcs  on  the  chart  draw  straight  lines  and  mark 
the  intercepts  H±  H2,  74 12.  .  .  . 

138.  Upon  completing  the  chart,  the  surface 
of  the  cam  is  drawn  as  in  Fig.  61,  with  a  diam- 


eter  E'  Tr  = 


=  16.12.     The  width  CN  of 


the  cylinder  may  be  taken  equal  to  the  chord 
A  V  of  the  arc  of  swing  of  the  follower  pin,  plus 
twice  the  diameter  of  the  pin. 

139.  The  simplest  general   plan   for    trans- 
ferring the  cam    chart  to   the   surface    of  the 
cam  is  to  consider  the  chart  lines  to  be  on  a 
strip  of  paper,  and  that  this  paper  is  simply 
wound  around  the    cylindrical   surface   of  .the 
cam,  starting  the  point  G  of  the  chart  at  G  on 
the  center  line  of  the  cam.    G  on  the  chart  is 
midway  between  J  and  D.     Then  the  points 
#2,  Iz  ...  of  the  base  curve  in  Fig.  60  will  fall 
at  H2,  72,  in  Fig.  61,  giving  the  surface  guide 
curve  for  the  center  of  the  cutting  tool. 

140.  The  detail  necessary    to   actually   lo- 
cate the  points  H2)  1%  in  Fig.  61  is  accomplished 
by  projecting  J  to  J'  and  laying  off   the  as- 


ELEMENTARY   CAMS 


signed  120°  divisions,  and  also  the  subdivisions  from  this  latter 
point.  The  120°  divisions  are  shown  at  Mf,  Q',  J';  the  equal 
subdivisions  at  H3 I3.  .  .  .  From  these  latter  points,  lines  are  pro- 
jected to  the  front  view  and  the  lengths  #4  H2,  74  72  are  transferred 


FIG.  61.— CYLINDRICAL  CAM  WITH  SWINGING  FOLLOWER  DRAWN  FROM  CHART 

from  Fig.  60.  To  find  the  point  of  tangency  at  E,  make  K4  EI 
of  Fig.  60  equal  to  K3  E'  of  Fig.  61,  then  draw  El  E  in  Fig.  60 
and  lay  off  this  distance  from  the  center  line  G  Y  in  Fig.  61,  thus 
giving  the  point  E.  To  find  the  point  of  tangency  at  M,  lay  off 
at  M'  M3  a  distance  equal  to  the  chart  distance  from  M2  to  M 
in  Fig.  60  and  project  M3  of  Fig.  61  to  M . 


SECTION   IV.— TIMING   AND   INTERFERENCE   OF  CAMS 

141.  In  machines  where  two  or  more  cams  are  employed  it  is 
generally  necessary  to  lay  down  a  preliminary  diagram  showing 
the  relative  times  of  starting  and  stopping  of  the  several  cams,  in 
order  to  be  assured  that  the  various  operations  will  take  place  in 
proper  sequence  and  at  proper  intervals.     The  same  preliminary 
diagram  is  also  used  to  avoid  interference  and  to  make  clearance 
allowances  for  follower  rods  whose  paths  cross  each  other. 

142.  PROBLEM  17.     CAM  TIMING  AND  INTERFERENCE.     Required 
two  cams  that  will  operate  the  follower  rods  A  and  E,  Fig.  62,  lying 
in  the  same  plane,  so  that: 

(a)  Rod  A  shall  move  16  units  to  D,  dwell  for  30°,  return  8  units 
to  B  and  again  dwell  30°,  all  to 

take  place  in  180°  turn  of  the 
cam.  The  cam  to  produce  the 
same  motions  in  the  second  180° 
but  in  reverse  order. 

(b)  Rod  E  shall  cross   path 
of  rod  A  and  move  4  units  be- 
yond it  and  back  again  during 
the  time  that  rod  A  is  moving 
from  D  to  B  to  D. 

All  motions  to  be  on  the 
crank  curve  with  maximum  pres- 
sure angles  of  40°. 

143.  Before    taking    up    the 
solution  of  this  problem   in   de- 
tail it  should  be  noted :  1st,  that  FIG.  62. 
any  convenient  type  of  cam  may 

be  used  in  problems  of  this  kind ; 

2d,  that  usually  only  general  motions  of  followers  or  objects  are  given 
in  the  preliminary  data,  as  above,  and  that  the  cam  designer  must 
supply  data  and  restate  the  problem  in  terms  of  angles  for  each  of 
the  movements  after  studying  the  preliminary  data  with  the  aid  of 
a  timing  diagram. 

144.  The  first  step  leading  to  a  restatement  of  the  problem  is  to 
determine  the  number  of  degrees  in  which  rod  A  may  move  the 

75 


PROBLEM  17,  PRELIMINARY  LAYOUT 
OF  DATA  FOR  PROBLEM  IN  CAM  INTER- 
FERENCE 


76 


ELEMENTARY   CAMS 


16  units,  and  also  the  number  of  degrees  in  which  it  may  move 
the  8  units  in  order  that  the  pressure  angle  will  be  40°  in  both  cases. 
Since  there  are  two  30°  dwells  in  the  first  180°  there  will  be  120° 

left  for  the  two  motions    of   which   the   first 
cT  ^  IQ 

motion  will  require  ^r  of  120°  or  80°,  and  the 

second,  40°.     The  length  of  chart   for  cam  A 

360 
may  now  be  computed    as    16  X  1.87  X  -TTTT 

=  134.6  and  laid  off  as  at  A  AI,  Fig.  63.  The 
height  of  the  chart  A  D  is  16  units.  The 
chart  is  next  divided  into  degrees  of  any  con- 
venient unit,  0,  10°,  20°  .  .  .  being  used  in 
this  case.  For  the  present  the  base  line  may 
be  made  up  of  a  series  of  straight  lines  as  at  A 

~r\       T~\     T\       ~r\     ~D 

L/i,  L/i  L/2)  JJ2  r>i.   .   .   . 

145.  The    amount    of    clearance    between 
the  moving  arms  must  now  be  decided  upon. 
Let  it  be  the  designer's  judgment  that  the  end 
of  the  follower  rod  E  should  lie  at  rest  1  unit 
to  the  left  of  rod  A  as  shown  in  Fig.  64,  and 
that  rod  E  should  not  begin  to  move  until 
the  rod  A  is  one  unit  out  of  the  way.     Then 
A  will  be  atO,  Fig.  64,  moving  down,  when 
E  starts,  assuming  the  rod  E  to  be  3  units 
wide  and  that  it  is  so  placed  that  its  top  edge 
is  one  unit  below  D.     The  point  C  is  then  5 
units  from  the  top  of  the  stroke  and  if  this 
distance  is  laid  off  in  Fig.  63,  as  shown,  the 
line  C  Ci  is  obtained  cutting  the  crank  curve, 
which  should  now  be  drawn  at  C.     C  is  at 
the  133°  point  and  this,  then,  is  the  time  when 
the  follower  E  should  start  to  move. 

146.  The  total  motion  for  rod  E  is  4  +  5  + 
1  =  10  units,  assuming  width  of  rod  A  to  be  5 
units.     The  time  during  which  this  motion  can 
take  place,  outward,  is  180°  -  133°  =  47°  as 

represented  at  EI  E2,  Fig.  63.  If  the  crank  curve  EI  F  is  now  drawn  it 
will  be  intersected  by  the  one-unit  clearance  line  GiGatG  which  rep- 
resents, in  this  case,  a  rotation  of  approximately  11°  of  the  cam  that 
drives  rod  E.  The  total  clearance  for  the  two  rods  which  cross  each 


i 

b 

/ 

^ 

K 

/ 

I 

* 

5? 
V 

9 

V 

r  V" 

o 

\ 

ea 

/ 

3  '<? 

&, 

/ 

•D       '—  ' 

v  

H 

v\  —  - 

o 

«-JHi 

3 

v 

/ 

% 

0      ^S 

^J 

TT/p- 

_^__ 

p=H 

H 

1 

-V  ^ 

ol 

S  3 

^ 

~ 

f^ 

| 

(- 

1 

H    -I/ 

io5 

~ 

,cT 

\ 

g' 

\ 

\ 

\ 

fe 

\ 

\ 

\ 

, 

£ 

\ 

^ 

Ra 

«  1 

I  > 

[<—  01  —  > 

1 

FIG.  63.— PROBLEM  17, 
TIMING  DIAGRAJM  FOR 
AVOIDING  INTERFER- 
ENCE OF  CAMS 


TIMING   AND    INTERFERENCE    OF   CAMS 


77 


other's  paths  is  now  found  to  be  3°  for  cam  follower  A  and  11°  for  cam 
follower  E,  or  14°  of  the  machine  cycle.  These  clearances  are  in- 
dicated in  Fig.  63.  If  it  is  the  judgment  of  the  designer  that  errors 
in  cutting  keyways  and  in  assembling,  and  that  the  wear  of  the  parts 
will  fall  within  these  limits,  the  cams  may  now  be  drawn. 

147.  The  cam  chart  for  cam  E  was  made  the  same  length  as 


FIG.  64. — PROBLEM  17,  DESIGN  FOR  DEFINITE  TIMING  AND  NON-INTERFERENCE  OF 
CAMS  OPERATING  IN  SAME  PLANE 

the  chart  for  cam  A  in  order  to  make  clearance  allowance.     The 
true  length  of  this  chart,  for  a  40°  pressure  angle,  would  be: 

OCA 

10  X  1.87  X  -47-  =  143.2  units, 

instead  of  134.6  as  now  drawn.     If  an  exact  clearance  allowance 
in  degrees  were  required,  it  would  be  necessary  to  redraw  the  crank 

47 
curve  EI F,  making  the  distance  EiE2  equal  to  ~™  of  143.2.     It  is  now 

47 

of  134.6.     With  a  new  and  exact  drawing  the  crank  curve  EI  G 


78 


ELEMENTARY   CAMS 


would  not  rise  quite  so  rapidly  and  the  intercept  at  G  would  show  a 
small  fraction  over  the  11°  taken  above.  In  some  problems  where  the 
lengths  of  the  true  charts  differ  considerably  it  may  be  necessary  to 
redraw  this  part  of  the  base  curve  to  be  sufficiently  accurate  in 
obtaining  the  clearance  in  degrees. 

148.  The  radius  of  the  pitch  circle  for  the  cam  operating  rod  A 


•11  u    134-6 
will  be 


21.4  units  as  drawn  at  HI,  Fig.   64.     The  pitch 


surface  of  the  cam  and  the  working  surface  are  drawn  in  the  same 
way  as  the  ordinary  radial  cams  in  previous  problems.  The  length 
of  the  rod  AI  A  may  be  assumed. 

The  radius  for  the  pitch  circle  for  the  cam  operating  rod  E  will  be 


143  2 

-T    r  =  22-8  and  this  is  laid  off  at 


The  location  of  M  and  the 


LL 

,   « 

i  1^       i      >l      ' 

C 

| 

1                 1 

1             /~f             I 

L        J 

| 

length  of  the  rod  N  E  will  either  enter  into  the  layout  of  the  frame- 
work of  the  machine  in  a  practical  problem,  or  will  be  determined 

by  the  framework  if  pre- 
viously laid  out.  In  the 
present  case  it  will  only 
be  necessary,  in  deter- 
mining the  length  of  rod 
N  E  and  the  position 
of  M,  to  make  certain 
that  the  shafts  M  and  H 

FIG.  65.-PBOBLEM  I7a,  DIAGRAM  SHOWING  AFPLI-       are  Sufficiently  far  apart 

CATION  OF  DATA  to  keep  the  cams  from 

striking  when  turning. 

149.  LOCATION  OF  KEYWAYS.      It  is  important  to  locate  the 
keyway  exactly  by  giving  its  position  in  degrees  so  as  not  to  destroy 
the  clearance  values  already  made.     Since  the  working  surfaces  of 
cams  frequently  approach  close  to  the  hub  or  shaft  it  is  a  good 
plan  to  place  the  keyway  at  the  center  of  the  longest  lobe  of  the 
cam,  as  illustrated  in  both  cams. 

150.  EXERCISE  PROBLEM  17a.     Assume  a  stack  of  blocks  at  A, 
Fig.  65.     Required  that  the  bottom  block  shall  be  delivered  with 
one  stroke  at  C,  the  next  block  at  D,  being  moved  first  to  B  and 
then  to  D,  the  next  block  at  C,  the  next  at  D,  etc.     Let  the  sizes  of 
the  blocks  and  the  distances  they  must  be  moved  be  as  shown  in 
Fig.  65.     Lay  out  cam  mechanism  to  secure  this  result,  keeping 
the  maximum  pressure  angle  at  30°. 


SECTION   V.— CAMS  FOR  REPRODUCING   GIVEN    CURVES 

OR  FIGURES 

151.  PROBLEM  18.     CAM  MECHANISM  FOR  DRAWING  AN  ELLIPSE. 
Required  a  cam  mechanism  that  will  reproduce  the  ellipse  AC  B  D 
in  Fig.  66,  the  marking  point  to  move 'slowly  at  the  extremities  A 
and  B  of  the  major  axis  and  rapidly  at  C  and  D,  the  rate  of  increase 
and  decrease  of  velocity  being  uniform. 

152.  Divide  A  C  into  three  parts  which  are  to  each  other  as  1, 
3,  and  5;  C  B  into  three  parts  which  are  as  5,  3,  and  1  ...  in  order 
that  the  marking  point  shall  move  through  increasing  spaces  in 
equal  times  in  moving  from  A  to  (7.  ...  For  greater  accuracy  A  C 
would  be  divided  into  a  greater  number  of  parts. 

153.  In  devising  the  mechanism  assume  that  the  marking  point 
shall  be  at  the  end  of  a  rod  which  shall  be  controlled  by  two  com- 
ponent motions  that  are  horizontal  and  vertical,  or  nearly  so.     This 
suggests  the  rod  A  E  F,  with  marking  point  at  A,  with  horizontal 
motion  supplied  from  a  bent  rocker  attached  at  F  and  with  vertical 
motion  supplied  from  a  reversing  straight  arm  rocker  L  K  J,  attached 
through  a  link  E  J  at  the  point  E.     The  lengths  of  the  links  and  of 
the  arms  of  the  rockers,  and  the  positions  of  the  fixed  centers  of  the 
rockers  will  have  to  be  assumed,  the  lengths  of  the  arms  and  links 
being  such  that  none  of  them  will  have  to  swing  through  more  than 
60°.     With  more  than  60°  swing  the  angle  between  an  arm  and  a  link 
is  liable  to  become  too  acute  for  smooth  running.     Where  rocker 
arms  are  connected  to  links  the  ends  of  the  rocker  should,  in  general, 
swing  equal  distances  above  and  below  the  center  line  of  the  link's 
motion,  as  for  example,  the  points  F  and  6  on  the  arc  of  swing  of  F 
should  be  as  much  above  the  line  A  M  as  the  point  3  is  below.     Also 
the  arc  3  J  9  should  swing  equally  on  each  side  of  J  T  in  order  to 
secure  best  average  pressure  angles  for  the  mechanism. 

154.  Let  each  of  the  rocker  arms  be  assumed  to  be  controlled  by 
single-acting  radial  cams.     The  center  of  roller  H  will  be  required 
to  swing  on  an  arc  6  H  which,  continued,  passes  through  M .     This 
gives  small  pressure  angles  while  A  is  traveling  to  B,  especially  when 
A  is  at  C  and  is  moving  fastest.     It  gives  large  pressure  angle,  how- 
ever, while  A  is  traveling  from  B  to  D  to  A.     If  A  is  assumed  to  do 
heavy  work  along  A  C  B  and  to  run  light  along  B  D  A  this  is  the 

79 


80  ELEMENTARY   CAMS 

better  arrangement.  If  A  did  the  same  work  on  both  strokes  it 
would  be  better  to  place  the  rocker  arm  G  H  so  that  H  and  6  rested 
on  a  radial  line.  The  center  of  roller  L  will  be  assumed  to  travel 
on  an  arc  whose  extremities  are  on  a  radial  line,  or  nearly  so. 

With  A  F  as  a  radius  and  A,  1,  2  ...  as  centers,  strike  short 
arcs  intersecting  F  6  at  F,  1,  2  .  .  .  numbering  the  arcs  as  soon  as 
drawn  to  avoid  confusion  later  on.  Lay  off  points  on  H  6  corre- 
sponding to  those  on  F  6. 

155.  Inasmuch  as  the  point  H  does  not  move  in  accordance  with 
the  law  of  any  of  the  base  curves  no  precise  computation  can  be 
made  for  the  size  of  the  pitch  circle  for  any  given  pressure  angle 
and  it  may  be  omitted.     Instead,  a  minimum  radius  M  H  of  the 
pitch  surface  may  be  assumed.     If  it  is  desired  to  control  the  pres- 
sure angles  it  may  be  done  by  first  constructing  the  pitch  surface, 
H  V  W,  and  then  measuring  the  angles  at  the  construction  points. 
Some  of  these  are  shown  in  Fig.  66,  at  H,  3,  6,  and  8,  and  are  20°, 
— 12°,  48°,  and  57°,  respectively.     If  these  angles  should  prove  un- 
satisfactory a  larger  pitch  circle,  or  a  differently  proportioned  rocker, 
may  be  used.     Or,  an  approximate  computation  for  radius  of  pitch 
circle  by  the  method  which  is  explained  to  advantage  in  connection 
with  the  next  problem,  paragraphs  164  and  165,  may  be  used. 

156.  To  construct  the  second  cam,  take  the  distance  A  E  as  a 
radius  and  A,  1,2  .  .  .  as  centers  and  mark  the  points  E,  1,  2.  .  .  . 
Again,  with  the  latter  points  as  centers  and  E  J  as  a  radius,  mark 
the   points  J,  1,  2  .  .  .  and  transfer  these  to  L,  1,  2.  .  .  .  With 
the  latter  points  marked,  the  pitch  surface  of  the  second  cam,  P  Q  R, 
is  constructed  in  the  same  way  as  was  the  first  cam. 

The  angle  between  the  keyways,  marked  39J^°  in  Fig.  66,  must 
be  carefully  measured  and  shown  on  the  drawing. 

157.  EXERCISE  PROBLEM  18a.     Required  a  cam  mechanism  that 
will  draw  the  numeral  8,  the  marking  point  moving  with  uniform 
velocity. 

158.  PROBLEM  19.     CAMS  FOR  REPRODUCTION  OF  HANDWRITING. 
Required  a  cam  mechanism  to  reproduce  the  script  letters  S  t  e. 

159.  The  first  step  in  the  solution  of  this  problem  is  to  write  the 
letters  carefully,  for  if  the  machine  is  properly  designed  it  will  re- 
produce the  copy  exactly  as  written.     The  copy  is  written  at  A  in' 
Fig.  67. 

160.  The  next  step  is  to  decide  on  the  kind  of  mechanism  and 
the  type  of  cams  to  be  used,  for  the  problem  may  be  solved  by  a 
number  of  different  combinations.     The  mechanism  for  this  problem 


FIG.  66. — PROBLEM  18,  CAM  FOR  DRAWING  AN  ELLIPSE 


+     -f-     f 

PIG.  67.— PROBLEM  19,  CAMS  FOR  REPRODUCING  SCRIPT  LETTERS,  ETC. 


CAMS   FOR  REPRODUCING   GIVEN   CURVES   OR   FIGURES       83 


will  consist  of  two  radial  single-acting  cams 
mounted  on  one  shaft,  and  a  swinging  rocker 
arm  mounted  on  a  pivot  which  is  moved  forth 
and  back  on  a  radial  line  as  shown  in  Fig.  67. 
This  mechanical  combination  is  selected  for  this 
problem  because  it  involves  methods  of  construc- 
tion not  used  in  any  of  the  preceding  problems. 

161.  The  actual  work  of  construction  is  started 
by  marking  off  a  series  of  dots  along  the  lines  of 

|  the  entire  copy,  as  shown  at  A,  and  marked  from 
K  zero  to  64.  Inasmuch  as  there  is  some  latitude 
|  in  the  spacing,  and  consequently  in  the  number 
s  of  these  dots,  as  will  be  explained  presently,  it  is 
o  advisable  to  use  a  total  number  of  dots  whose 
o  least  factors  are  2  and  2,  2  and  3,  or  2  and  5. 
|  This  is  not  essential  but  it  will  facilitate  the  work 
^  later  on. 

162.  The  matter  of  placing  the  dots  is  per- 
il    haps  the  most  important  item  of  the  entire  prob- 
3     lem,  for  on  this  depends  the  size  of  the  roller  and 
g     smooth  action.   In  fact,  with  some  methods  of  spac- 
g     ing,  no  roller  can  be  used  at  all  and  a  sharp  V- 
£     edge  sliding  follower  will  have  to  be  used  if  true 
J}     reproduction  is  desired. 

The     basic     considerations     in    selecting    the 
£     points  are: 

First,  that  a  point  should  be  located  at  the 
extreme  right  and  extreme  left  of  each  right  and 
left  throw,  as  at  0  -  7,  7  -  16, 16  -20  .  .  .  in  Fig. 
67,  A,  and  at  the  top  and  bottom  of  each  swing, 
as  at  0  -  8,  8  -  13,  13  -  18  .  .  .;  and, 

Secondly,  that  the  marking  point  should  start 
slowly  and  come  to  rest  gradually  on  each  stroke, 
considering  both  of  the  component  directions  of 
its  motion  at  the  same  time.  On  account  of  this 
it  is  impossible  to  secure  ideal  conditions  at  all  times  and  com- 
promises must  frequently  be  made.  For  example,  the  component 
motions  of  the  marking  point  D  are:  First,  a  horizontal  one  due 
to  Cam  No.  1 ;  and  secondly,  a  vertical  curvilinear  one  due  to  Cam 
No.  2  and  the  rocker  arm  H  G  D.  The  intermediate  points  0-7  on  the 
upper  swing  of  the  letter  S  are  so  selected  as  to  give  increasing  and 


84  ELEMENTARY   CAMS 

decreasing  spaces  in  the  horizontal  projections  on  D  E,  and  the 
same  points,  together  with  point  8,  are  selected  at  the  same  time  so 
as  to  give  increasing  and  decreasing  spaces  when  projected  onto  the 
arc  D  F.  Each  space  between  a  pair  of  adjacent  numbers  represents 
the  same  time  unit,  On  this  basis  the  entire  spacing  of  the  copy 
is  done. 

163.  With  each  of  the  points  in  the  group  at  A,  Fig.  67,  as  centers, 
and  with  a  radius,  D  G,  mark  very  carefully  the  corresponding  points 
on  G  L  in  group  B.     To  avoid  confusion  it  is  essential  here  to  adopt 
some  method  of  identifying  points  so  marked  for  later  reference. 
A  satisfactory  method  is  shown  at  B,  all  the  motions  to  the  right 
being  indicated  below,  and  the  motions  to  the  left,  above  G  L. 

164.  The  sizes  of  the  cams  are  to  be  next  computed.     To  do 
this  select  the  largest  horizontal  space  in  section  A.     This  is  found 
between  56  and  57  and  is  equal  to  .46  of  the  unit  of  length  that 
happened  to  be  selected  in  this  problem.     Assuming  that  the  marking 
point  moves  with  uniform  velocity  over  this  distance,  and  that  a 
pressure  angle  of  40°  is  suitable  in  this  instance  where  no  heavy 
work  is  done,  the  factor  of  1.19  is  taken  from  the  table  in  paragraph 
30.     Since  there  are  64  time  units  the  length  of  circumference  of 
pitch  circle  for  Cam  No.  1  will  be 

.46  X  1.19  X  64  =  35.03,  and  the  radius  5.58. 

165.  Before  calculating  the  size  of  Cam  No.  2  the  length  of  the 
rocker  arm  G  H  must  be  decided  upon  and  this  will  be  taken  in  this 
problem  at  5  units,  the  same  as  the  arm  G  D.     Then  the  total  swing 
of  the  follower  point  H  will  be  H  K,  equal  to  D  F,  and  the  greatest 
swing  in  any  one  direction  in  any  one  time  unit  will  be  during  the 
periods  10-11  and  61-62,  shown  at  A,  Fig.  67,  both  equal  to  .48  units. 
Making  the  same  computation  as  for  Cam  No.  1, 

.48  X  1.19  X  64 


3.14  X  2 


=  5.82 


equals  the  pitch  radius  of  Cam  No.  2. 

166.  The  position  of  the  cam  shaft  0  relatively  to  the  pivot 
arm  G  depends  on  what  is  desired  for  the  position  of  the  arc  H  K 
with  reference  to  the  cam  center.  If  it  is  desired  that  the  points 
H  and  K  shall  be  on  a  radial  line  from  the  center  of  the  cam,  which 
gives  best  practical  average  results  for  both  in  and  out  strokes, 
proceed  as  follows:  Draw  chord  D  F  at  A  in  Fig.  67;  bisect  it  at 
J  and  measure  distance  G  J  which  is  4.93  units.  Then  the  distance 


CAMS   FOR   REPRODUCING   GIVEN   CURVES   OR   FIGURES         85 

G  0  will  be  the  hypothenuse  of  a  right  angle  triangle  of  which  one 
side  is  4.93  and  the  other  5.82.  This  may  be  separately  drawn  and 
the  length  of  the  hypothenuse  found  graphically  or  it  may  be 
figured  as  follows: 


G  0  =      5.822  +  4.932  =  7.63. 

167.  The  pitch  circles  for  both  cams  may  be  taken  in  problems 
of  this  kind  to  pass  through  the  midpoint  of  the  total  travel.     Then 
0  M  is  the  radius  of  the  pitch  circle  of  Cam  No.  1  and  N  P  the 
total  range  of  travel  of  the  roller  center;   and  0  Ji  is  the  radius  of 
the  pitch  circle  of  Cam  No.  2  and  H  K  the  total  range  of  travel  of 
the  roller  center  relatively  to  G. 

168.  To  find  points  on  the  pitch  surface  of  the  cams  proceed  in 
the  usual  way  for  Cam  No.  1,  by  dividing  the  circle  whose  radius  is 
0  N  into  as  many  equal  parts  as  there  were  dots  on  construction 
points  at  A.     Draw  radial  lines,  and  on  these  lay  off  the  distances 
secured  from  B  in  Fig.  67 ;    for  example,  the  distance  3  NI  is  laid 
out  equal  to  G  3.     The  point  NI,  and  other  points  secured  in  similar 
manner,  will  lie  on  the  pitch  surface  of  Cam  No.  1. 

169.  The  construction  of  the  pitch  surface  for  Cam  No.  2  is 
different  from  that  of  Cam  No.  1,  and  is  different  also  from  anything 
done  in  the  preceding  problems.     In  this  case  the  resultant  motion 
of  the  arm  G  H  is  made  up  of  rectilinear  translation  and  rotation  and 
both  components  must  be  considered  in  laying  out  the  pitch  surface, 
for  example,  as  follows :    With  G  H  as  a  radius  and  point  4  of  B  as 
a  center  draw  an  arc  intersecting  the  horizontal  line  through  H  at  4- 
Then  when  G  is  moved  to  4  by  Cam  No.  1,  H  would  be  at  4  if  the 
rectilinear  component  motion  due  to  cam  No.  1  were  the  only  one 
acting.     During  the  period  represented  by  G  4,  however,  Cam  No.  2 
must  move  the  rocker  arm  through  an  arc  Q  4,  shown  at  A,  and  this 
arc  must  now  be  laid  off  at  4  R-    The  point  R  is  then  revolved  to 
its  proper  position  at  T  as  follows :    Divide  the  circle  0  G  into 
sixty-four  equal  parts.     This  is  readily  done  in  this  problem  because 
G  is  taken  on  the  same  radial  line  with  N  and  the  radial  divisions 
already  made  on  the  circle  having  0  N  for  a  radius  need  only  be 
extended.     Lay  off  the  distance  G  4  at  4  S.     With  S  as  a  center  and 
G  H  as  a  radius  draw  the  arc  4  T.    Then  T  will  be  a  point  on  the 
pitch  surface  of  Cam  No.  2. 

170.  Having  determined  the  pitch  surfaces  of  the  two  cams  the 
largest  possible  roller  for  each  is  found  by  searching  for  the  shortest 
radius  of  curvature  on  the  working  side  of  each  pitch  surface.     For 


86 


ELEMENTARY   CAMS 


Cam  No.  1  the  size  of  the  largest  roller  that  can  be  used  is  that  of 
the  circle  whose  center  is  at  U;  and  for  Cam  No.  2  it  is  that  of  the 
circle  whose  center  is  at  V.  In  order  to  avoid  sharp  edges  on  the 
cams,  rollers  slightly  smaller  than  these  circles  will  be  used. 

171.  For  assembling  the  cams  the  angles  between  them  and  the 
angles  for  the  keyways  should  be  carefully  measured  and  placed  on 
the  drawing  as  shown  in  Fig.  67. 

A  front  view  showing  the  elevations  of  the  cams,  lever  arm,  slide, 
and  plate  is  given  in  Fig.  68. 

172.  METHOD  OF  SUBDIVIDING  CIRCLES  INTO  ANY  DESIRED  NUM- 
BER OF  EQUAL  PARTS.     The  matter  of  subdividing  the  circle  having 


C   M 


FIG.  69.— METHOD  OF  SUBDIVIDING  CIRCLES  INTO  ANY  DESIRED  NUMBER  OF  EQUAL  ARCS 

radius  0  N,  Fig.  67,  into  sixty-four  equal  parts  was  a  simple  matter 
of  subdivisions.  If  it  is  required  to  divide  the  circle  into  eighty-seven 
equal  parts  the  work  is  just  as  simple  if  a  proper  start  is  made  as 
follows:  Let  it  be  required  that  the  circle  B  D,  Fig.  69,  be  divided 
into  eighty-seven  equal  parts.  Find  the  number  next  lower  than 
eighty-seven  whose  least  factors  are  2  X  2,  2  X  3,  or  2  X  5.  Such  a 
number  is  80.  Assume  that  the  circle  is  6  inches  in  diameter;  then 
the  circumference  is  18.84  inches  and  %7  of  this  is  1.516  inches, 
which  is  laid  off  to  scale  on  the  tangent  at  B  F.  With  a  pair  of  small 
dividers,  set  to  any  convenient  small  measuring  unit,  step  off  divisions 


CAMS   FOR   REPRODUCING   GIVEN   CURVES   OR   FIGURES       87 

from  F  to  the  next  step  beyond  B.  Assume  that  there  are  11  steps 
from  F  to  G,  then  go  forward  11  steps  on  the  arc  to  K.  Divide  the 
large  part  of  the  circle  K  D  B  into  eighty  parts  by  the  process  of  sub- 
division with  the  dividers  as  indicated  by  the  divided  angles  80,  40, 
20,  4,  2,  and  1,  in  Fig.  69.  Then  B  H  is  %>  of  K  D  B,  or  %,  of 
the  entire  circle,  and  the  length  B  H  will  go  exactly  seven  times  into 
the  arc  B  K.  In  this  work  nothing  is  said  of  the  use  of  a  protractor 
for  laying  off  a  large  number  of  small  subdivisions  on  a  circle,  al- 
though it  may  be  used.  The  process  of  subdivision,  however,  always 
using  the  small  dividers,  gives  automatically  remarkably  accurate 
results. 


SECTION   VI.— ADVANCED   GROUP   OF  BASE   CURVES 

FOR  CAMS 

173.  THE  PREVIOUS  SECTIONS  OF  THE  BOOK  have  dealt  with  the 
simpler  base  curves  which  are  in  common  use,  and  with  their  ele- 
mentary application  to  various  types  of  cams.     In  the  present  sec- 
tion the  simpler  forms  of  base  curves  are  further  considered,  other 
forms  are  treated,  and  new  ones  are  proposed;    all  are  brought 
together  for  comparisons. 

174.  COMPLETE  LIST  OF  BASE  CURVES.     The  base  curves  which 
have  been  used  in  the  previous  sections  are: 

Straight  Line,  Figs.  22  and  78. 
Straight-Line  Combination,  Figs.  23  and  82. 
Crank  Curve,  Figs.  24  and  86. 
Parabola,  Figs.  25  and  90. 
Elliptical  Curve,  Figs.  26  and  102. 

Other  base  curves  which  will  be  considered  in  following  para- 
graphs are: 

All-Logarithmic  Curve,  Fig.  70. 
Logarithmic-Combination  Curve,  Fig.  74. 
Tangential  Curve,  Case  1,  Fig.  94. 
Circular  Curve,  Case  1,  Fig.  98. 
Cube  Curve,  Case  1,  Fig.  106. 
Circular  Curve,  Case  2,  Fig.  110. 
Cube  Curve,  Case  2,  Fig.  114. 
Tangential  Curve,  Case  2,  Fig.  118. 

175.  COMPARISON  OF  BASE  CURVES,  THEIR  APPLICATIONS,  AND 
THEIR  CHARACTERISTIC  MOTIONS.     Figs.  70  to  121  illustrate: 

(1)  The  forms  of  each  of  the  base  curves,  Column  1. 

(2)  The  form  and  true  relative  size  of  cam,  all  having  the  same 
data,  Column  2. 

(3)  The  velocity  diagram  for  each  cam,  Column  3. 

(4)  The  acceleration  diagram  for  each  cam,  Column,  4. 

176.  THE  DATA  FOR  ALL  OF  THE  CAMS  and  diagrams  illustrated  in 
Figs.  70  to  121  are  as  follows: 

(a)  The  follower  to  rise  1  unit  in  60°  turn  of  the  cam, 

(b)  "         "        "  fall  1  unit  in  60°     "    "    "      " 

(c)  The  follower  to  remain  at  rest  for  240°  turn  of  the  cam, 

(d)  ' '   maximum  pressure  angle  to  be  30°. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     89 

177.  ALL  OF  THE  CAM  CHARTS  illustrated  in  Column  1,  Figs.  70 
to  121,  include  only  the  first  item  in  the  above  data  and  they  show, 
therefore,  only  one-sixth  of  their  full  length.     In  Column  2  the  entire 
cam  is  shown  in  each  case,  and  it  is  drawn  to  one-third  of  the  scale 
used  for  the  chart  in  Column  1. 

178.  VELOCITY  AND   ACCELERATION  'DIAGRAMS   SHOWING   CHAR- 
ACTERISTIC   ACTION    OF    CAMS    HAVING    DIFFERENT    FORMS    OF    BASE 

CURVES.  All  of  the  diagrams  in  Column  3,  Figs.  70  to  121,  show  the 
velocity  given  to  the  follower  by  the  cam  at  every  instant  during 
the  follower  stroke.  In  each  case  the  length  of  the  diagram  A  C 
represents  the  time  required  by  the  cam  to  turn  through  60°,  the 
cam  shaft  being  assumed  to  be  turning  with  uniform  angular  velocity. 
The  numbered  scale  on  each  diagram  shows  the  relative  velocity 
given  by  each  cam  at  any  phase  of  the  stroke. 

179.  All  of  the  diagrams  given  in  Column  4  show  the  acceler- 
ation given  to  the  follower  by  the  different  cams.     These  diagrams 
have  a  special  interest  when  it  is  remembered  that  force  =  mass 
X  acceleration,  and  if  the  mass  is  the  same  in  all  cases  the  ordinates 
of  the  diagrams  represent  the  forces  necessary  to  move  the  follower 
at  any  instant.     A  diagram  with  a  distinctively  long  ordinate  indi- 
cates that  the  cam  will  "run  hard  "  at  the  phase  where  the  long 
ordinate  occurs.     The  scale  numbers  shown  on  the  diagrams  are 
based  on  the  uniform  acceleration  given  by  the  parabola  cam  as 
shown  in  Fig.  93. 

180.  THE     CHARACTERISTIC    ACTIONS    OF    DIFFERENT    CAMS    built 

from  the  various  base  curves  will  be  considered,  in  order,  in  the  fol- 
lowing paragraphs. 

181.  THE  ALL-LOGARITHMIC  CURVE,  Fig.  70,  gives  the  smallest 
possible  cam  for  a  given  pressure  angle.     It  differs  from  all  other  cam 
curves  in  that  it  gives  the  maximum  pressure  angle  all  the  time  that 
the  follower  is  moving,  whereas  the  others  give  a  maximum  pressure 
angle  for  an  instant  only.     One  of  the  disadvantages  of  the  all- 
logarithmic  cam  is  that  it  causes  the  follower  to  attain  nearly  its 
full  velocity  instantaneously,  and  causes  it  to  come  to  rest  in  a 
similar  manner,  thus  giving  a  shock  at  the  beginning  and  end  of  the 
stroke.     This  gives  excessively  large  acceleration  and  retardation 
at  the  ends  of  the  stroke  and  causes  the  cam  to  "  pound  "  or  "  run 
hard  "  at  these  phases  of  its  action.     Another  disadvantage  is  that  a 
roller  cannot  be  used  with  it  because  the  pitch  surface  has  a  sharp 
edge,  or  angle,  on  the  working  side  as  shown  at  C,  Fig.  71.     The  rea- 


90 


CAMS 


COMPARISONS  OF  CAMS  FOR  DIFFERENT 


-  COLUMN  1 

CAM  CHARTS  AND  BASE  CURVES 
TOR  ONE-SIXTH  OF. CAM 


COLUMN  2 
RELATIVE  SIZES  or  CAMF' 


FlQ.  70.     ALLrLOGARITHMIC    CURTS. 


Etfi.  74.     LOGARITHMIC-COMBINATION    OUHVB.  Pitch  Surface 


fFiQ.82.  STRAIGHT -LINK  COMBINATION  CURVE. 


Ho.  86.   CRANK  CURVB 


Pi».  90.  PARABOUL 


9i.  TANGENTIAL  BASK  CURTB,  CASK  1 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


91 


BASE  CURVES,  ALL  HAVING  SAME  DATA 


COLUMN  3 
VELOCITY  DIAGRAMS 


COLUMN  4 

ACCELERATION 

DIAGRAMS 


92 


CAMS 


COMPARISONS  OF  CAMS  FOR  DIFFERENT 


COLUMN  1 

CAM  CHARTS  AND  BASE  CURVES 
FOR  ONE-SIXTH  or  CAM 


Curve 


COLUMN  2 

RELATIVE  SIZES 

OF  CAMS 


FIG.  98.— CIRCULAR  CURVE,  CASE  1 

\n 


Pitch  Line 


\ 


£j/^ 


8 

&& 


-3.95- 


JK 

^FG:FC::7:4 

FIG.  102.  —  ELLIPTICAL  CURVE      FIG. 


-3.-7S- 


PIG.  106.—  CUBE  CURVE,  CASE  1    FIG.  107. 


L  — 


T* 

FIG.  110.— CIRCULAR  CURVE,  CASE  2        FIG.  111. 


FIG.  114.-  CUBE  CURVE,  CASE  2 


— TANGENTIAL  BASE  CURVE, 
CASE  2 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


93 


BASE  CURVES,  ALL  HAVING  SAME  DATA— Continued 

COLUMN  3  COLUMN  4 

VELOCITY  DIAGRAMS      ACCELERATION  DIAGRAMS 


FIG.  120. 


94 


CAMS 


son  why  a  roller  cannot  be  used  under  these  conditions  is  explained 
in  paragraph  59,  page  37.  The  construction  of  the  all-logarithmic 
cam  is  explained  in  the  following  paragraphs. 

182.  PROBLEM  20.     REQUIRED  AN  ALL-LOGARITHMIC  CAM  CAUSING  : 

(a)  The  follower  to  rise  1  unit  in  60°  turn  of  the  cam, 

(b)  "         "        "  fall  1     "    "  60°     "     "    "      " 

(c)  ' '         "        "  remain  stationary  for  240°  turn  of  the  cam, 

(d)  A  uniform  pressure  angle  of  30°. 

183.  A  BRIEF  GENERAL  ANALYSIS  for  the  method  of  procedure 
in  solving  an  all-logarithmic  cam  problem  is : 

(1)  To  construct  a  logarithmic  spiral  having  a  constant  normal 
angle  of  30°.     The  spiral  is  shown  at  B  H,  Fig.  122,  and  the  constant 


FIG.  70. — (Enlarged)  ALL-LOGARITHMIC 
BASE  CURVE 


FIG.  71— (Enlarged)  ALL- LOGARITH- 
MIC CAM 


angle  is  noted  at  J  D  K,  where  D  K  is  a  radial  line  and  D  J  a  line 
normal  to  the  curve. 

(2)  To  lay  out  the  assigned  working  angle  during  which  the 
follower  motion  takes  place,  on  a  piece  of  tracing  cloth  or  tracing 
paper,  as  at  6  in  Fig.  123. 

(3)  To  mark  on  each  leg  of  the  angle  a  scale  to  measure  the  fol- 
lower's motion,  as  at  0'  M  and  0'  N  in  Fig.  123. 

(4)  To  lay  the  tracing  cloth  represented  by  Fig.  123  over  the  loga- 
rithmic spiral  with  the  apex  Of  of  the  angle  always  at  the  pole  O  of 
the  spiral,  and  to  rotate  the  tracing  cloth  until  the  two  legs  of  the 
angle  cut  the  spiral  at  such  points  that  the  difference  in  length  of 
the  two  legs  is  equal  to  the  assigned  follower  motion.     This  is  illus- 
trated in  Fig.  122  where  the  shaded  area  represents  the  tracing-cloth 
with  the  assigned  angle  of  60°  shown  at  6,  while  0  C  minus  0  A  equals 
the  assigned  follower  motion  of  1  unit. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


95 


(5)  To  mark  the  included  part  of  the  logarithmic  spiral  A  C  and 
use  it  as  the  surface  of  the  cam  as  shown  at  A  C  in  Fig.  124. 

184.  THE  DETAIL  CONSTRUCTION  necessary  to  lay  out  the  all- 
logarithmic  cam  for  Problem  20  is  as  follows :  Construct  a  logarithmic 
spiral  with  a  constant  normal  angle  of  30°.  This  may  be  done 
mathematically  by  laying  off  computed  values  which  method  will  be 
taken  up  first,  or  it  may  be  done  graphically  as  will  be  explained  later. 


w 


A      1 

FIG.  122. — LOGARITHMIC  CURVE 
GIVING  CONSTANT  PRESSURE 
ANGLE  OF  30  DEGREES 


O' 


FIG.  123. — ASSIGNED  WORKING 
ANGLE,  TO  BE  DRAWN  ON 
TRACING  CLOTH 


In  the  mathematical  method  the  first  step  is  to  solve  the  following 
equation: 

r'_  _  ^Q  0.4343  J|Q  6  tan  (90°—  a) 

r 

where  a  is  the  assigned  press  ire  angle  and  b  is  a  unit  angle  taken  ai 
any  value  which  may  be  conveniently  used  later  in  starting  the 
drawing  of  the  spiral.  The  values  of  r  and  r'  are  shown  at  0  D  and 
0  H  respectively  in  Fig.  122.  The  angles  a  and  b  are  also  shown. 
A  convenient  angle  to  assume  for  6,  in  general,  is  60°  and  it  is  so 


taken  in  this  problem. 


Then  —  equals  the  number  whose  logarithm  is 


96    -  CAMS 


0.4343  X    ~  X  60°  X  tan  (90°  -  30°).     Solving,  the  value  of  the 

logarithm  is  0.2623  and  the  number  corresponding  to  this  is  1.83. 
Therefore, 


and  0  H,  Fig.  122,  is  made  1.83  times  0  D,  the  included  angle  being 
60°  in  accordance  with  the  above  assumption  for  b.  The  length  0  D 
may  be  taken  any  length  in  starting  the  construction  of  the  spiral. 
The  two  points  of  the  spiral  may  now  be  laid  down  as  at  D  and  H 
with  0  as  the  pole. 

185.  INTERMEDIATE  POINTS  ON  THE  LOGARITHMIC  SPIRAL  as  at  G 
may  be  found  by  bisecting  the  angle  D  0  H  and  making  0  G  a  mean 
proportional  between  0  D  and  0  H.    Then 

OD  :  OG  :    :  OG  :  OH 

If  0  D  is  taken  as  3  units,  then  0  G  =  V  3  X  (1.83  X  3)  =  4.06. 
To  find  points  on  the  spiral  at  closer  intervals  bisect  angle  DOG 
and  find  the  mean  proportion  0  7  which  is  equal  to  V  3  X  4.06  =  3.52. 
To  find  other  points  outside  of  a  given  angle,  such  as  at  5,  lay  off 
the  angle  DO  5  equal  to  angle  D  0  7  and  make  0  5  a  fourth  propor- 
tional to  0  7  and  0  D  as  follows: 

05  :  OD  :    :  OD  :  07 
Then  05  =  ^  =  2.58. 

If  points  are  desired  still  closer  together,  or  if  it  is  desired  to 
extend  the  spiral  in  either  direction,  it  may  be  done  by  the  above- 
described  processes,  or,  it  may  be  done  graphically  as  described  in 
paragraph  187. 

186.  The  next  detail  step  in  the  solution  of  Problem  20  is  to  draw 
an  angle  M  0  '  N,  Fig.  123,  on  tracing  cloth,  equal  to  the  assigned 
angle  of  60°  as  given  at  (a)  in  the  data,  and  lay  off  a  scale  on  each 
leg  of  the  angle  as  shown.     Then  lay  Fig.  123  over  Fig.  122,  0  ' 
always  at  0,  and  rotate  the  tracing  cloth  until  the  spiral  B  H  inter- 
cepts the  lines  0  '  M  and  0  '  N  at  such  points  that  0  '  C  '  minus 
0'  A'  equals  the  assigned  follower  motion  which  is  1  unit  as  stated 
at  (a)  in  Problem  20.     This  occurs  when  Fig.  123  is  at  the  position 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


97 


shown  by  the  section  lines  in  Fig.  122  where  0  A  equals  1.18  and  0  C 
equals  2.18.      The  intercepted  part  A  C  of  the  logarithmic  spiral 


FIG.  122. — (Duplicate)  LOGARITHMIC 
CURVE  GIVING  CONSTANT  PRESSURE 
ANGLE  OF  30  DEGREES 


FIG.  123. — (Duplicate)  ASSIGNED 
WORKING  ANGLE,  TO  BE  DRAWN 
ON  TRACING  CLOTH 


becomes  a  portion  of  the  cam  pitch  surface  as  shown  at  A  C  in 
Fig.  124  and  its  distance  from  the  center  of  rotation  of  the  cam  is 
the  same  as  the  distance  from  the  spiral  arc  to  the  pole  of  the  spiral. 


Fiajl24. — PROBLEM  20.     ALL-LOGARITHMIC  CAM  FOR  ASSIGNED  DATA 

Other  portions  of  the  cam  surface  are  found  in  a  similar  manner. 
As  shown  at  A,  E,  and  C,  in  Fig.  124,  the  pressure  angle  is  30°  at  all 
points. 


CAMS 


187.  Intermediate  points  on  the  logarithmic  spiral  may  be  found 
graphically,  instead  of  by  computation  as  given  in  paragraph  185, 
as  follows:  From  any  point  0  of  a  straight  line,  Fig.  125,  lay  off  0  D 
and  0  H  in  opposite  directions,  0  D  and  0  H  being  the  values  ob- 
tained by  computation  in  paragraph  184  and  shown  in  Fig.  122. 
At  0,  Fig.  125,  erect  a  perpendicular  line.  Find  the  midpoint  Oi 
on  the  line  D  H,  and  with  this  as  a  center  for  the  compass  draw  the 
semicircle  D  G  H.  Then  0  G  will  be  a  mean  proportional  between 
0  D  and  0  H  and  may  be  laid  out  as  the  ordinate  of  the  loga- 
rithmic spiral,  as  at  0  G,  Fig.  122,  where  0  G  bisects  the  angle  DO  H. 


s  o  'o,      Ko2  H 

Fio.  125. — GRAPHICAL  METHOD  FOR  FINDING  INTERMEDIATE  POINTS  ON  LOGARITHMIC 

CURVE 

To  find  a  fourth  proportional  graphically  proceed  as  follows: 
Lay  off  the  two  known  values,  0  D  and  0  H,  which  are  shown  in 
Fig.  122,  at  right  angles  to  each  other  as  shown  at  0  D'  and  0  H  in 
Fig.  125.  Find  the  point  62  on  0  H  that  is  equidistant  from  De 
and  H ,  and  with  this  as  a  center  draw  the  semicircle  H  D'  3,  giving 
the  length  0  3  as  the  fourth  proportional.  This  latter  distance  is 
laid  off  at  0  3  in  Fig.  122  where  the  angle  D  0  3  is  equal  to  angle 
DOH. 

188.  A  GRAPHICAL  METHOD  FOR  CONSTRUCTING  A  LOGARITHMIC 
SPIRAL  WHICH  HAS  A  GIVEN  CONSTANT  NORMAL  ANGLE  is  illustrated  in 

Fig.  126.  This  method,  referred  to  in  paragraph  184,  is  based  on 
the  following  theoretical  property  of  the  logarithmic  spiral,  namely, 
that  all  pairs  of  radiants  having  a  common  difference  embrace  equal 
lengths  of  arcs  on  the  spiral. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


99 


189.  The  principle  stated  in  the  previous  paragraph  may  be 
graphically  applied  only  approximately,  but  with  all  necessary  pre- 
cision, by  first  drawing  the  lines  M  P  and  P  N,  Fig.  126,  making  the 
desired  angle  with  each  other.  This  angle  will  be  30°  if  a  spiral  having 
a  constant  normal  angle  of  30°  is  required,  40°  if  a  constant  pressure 
angle  of  40°  is  required,  etc.  From  a  point  0,  where  the  vertical 
intercept  0  D  is  equal  to  about  the  estimated  short  radius  of  the  cam, 
draw  a  series  of  equidistant  vertical  lines  as  at  B,  C,  E,  etc.  With 
B  F  as  a  radius  and  0  as  a  center  draw  the  short  arc  1 ;  with  D  F  as  a 
radius  and  D  as  a  center  draw  arc  2.  The  intersection  of  arcs  1  and  2 
will  give  the  point  H  on  the  spiral.  Again,  with  C  G  as  a  radius  and 


FIG.  126. — GRAPHICAL  METHOD  FOR  CONSTRUCTING  A  LOGARITHMIC  CURVE  HAVING  A 
GIVEN  CONSTANT  NORMAL  ANGLE 

0  as  a  center  draw  arc  5;  and  with  F  G  (equal  D  F)  as  a  radius  and  H 
as  a  center  draw  arc  4-  The  intersection  of  arcs  3  and  4  will  give  a 
second  point  L  on  the  logarithmic  spiral.  It  will  now  be  noted  that 
the  two  pairs  of  radiants  H  0  -  D  0  and  L  0  -  H  0  have  a  common 
difference,  and  that  the  logarithmic  arcs  D  H  and  H  L  are  equal 
(approximately),  which  accords  with  the  general  principle  laid  down 
in  the  preceding  paragraph. 

190.  To  be  exact,  in  the  matter  of  the  graphical  construction  of 
the  logarithmic  spiral,  it  must  be  noted  that  it  is  the  chords  from  D  to 
H  and  from  H  to  L  that  are  equal  according  to  this  method  of  con- 
struction and  not  the  arcs  as  they  should  be  theoretically;  but  where 
the  vertical  construction  lines  are  taken  close  together  and  where  the 
distance  D  F  is,  therefore,  small,  the  error  in  the  curve  is  negligible. 


100 


CAMS 


In  the  present  case  the  ultimate  distance  0  R  when  drawn  with  aver- 
age care  to  a  scale  several  times  that  shown  in  Fig.  126,  varied  from 
the  computed  value  by  less  than  .01  inch.  The  part  of  the  curve 
from  D  to  Q  will  depart  from  theoretical  values  faster  than  the  part 
from  D  to  R,  due  to  the  sharper  curvature  of  D  Q,  but  the  effect 
of  this  may  be  overcome,  if  desired,  by  making  the  vertical  con- 
struction lines  to  the  right  of  0  D  closer  than  those  to  the  left  of  0  D. 

191.  THE     ALL-LOGARITHMIC     CAM    MAY    BE     CONSTRUCTED     BY    A 

PURELY  GRAPHICAL  METHOD,  and  without  any  mathematical  com- 
putation whatever.  In  Problem  20,  for  example,  it  would  only  be 
necessary  to  follow  the  directions  in  paragraphs  188  and  189,  making 
the  angle  a  of  Fig.  126  equal  to  30°  which  is  the  assigned  pressure 
angle  in  the  problem.  This  would  give  the  proper  logarithmic  curve 
identical  with  the  one  in  Fig.  122.  From  this  point  on,  the  direc- 
tions given  in  paragraph  186  apply.  If  a  pressure  angle  of  any  other 
size  were  desired,  say  45°,  the  angle  M  P  N  Fig.  126,  would  be 
made  45°. 

192.  EXERCISE  PROBLEM  20a.     REQUIRED  AN  ALL-LOGARITHMIC 
CAM  which  will  cause  a  follower  to : 

(a)  Rise  two  units  in  45°          turn  of  the  cam. 

(b)  Remain  stationary  for  135°     "    "    "     " 

(c)  Fall  two  units  in  45°  '"    "    "     " 

(d)  Remain  stationary  for  135°     "    il     ll     " 

(e)  The  constant  pressure  angle  to  be  30°. 

193.  A  LOGARITHMIC-COMBINATION  CAM  may  be  used  to  overcome 

the  disadvantages  (paragraph  181) 
of  the  all-logarithmic  cam  and  at 
the  same  time  to  sacrifice  very  little 
in  the  matter  of  increased  size.  This 
is  accomplished  by  substituting 
rounded  surfaces  for  the  angular 
surfaces  formed  by  the  all-logarith- 
mic curve.  When  the  rounded  sur- 
face thus  substituted  is  derived 
from  parabolic  base  arcs  the  best 
results  are  obtained.  A  cam  in 

FIG.  75.~"~~(EiiljLr£rG(l)   LOGARITHMIC—  i  •   ••     » i  •     i          i  i  •        i 

which  this  has  been  done  is  shown 
in  Fig.  75,  where  the  curves  A  Y  and 
Z  C  are  arcs  of  a  parabola  base  and  the  center  portion  7  Z  is  an 


ADVANCED  GROUP  OF  BASE  CURVES  FOB  CAMS     101 

arc  of  a  logarithmic  curve.  To  illustrate  an  actual  case,  a  prob- 
lem having  the  same  general  data  as  Problem  20  will  be  discussed 
in  the  following  paragraphs. 

194.  PROBLEM  21.     REQUIRED  A  LOGARITHMIC-COMBINATION  CAM 
causing  the  follower  to : 

(a)  Rise  1  unit  in  60°  turn  of  the  cam. 

(b)  Fall  1    "     "   "      "    "    "      " 

(c)  Remain  stationary  for  240°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  30°,  and  the  easing-off 
base  curves  to  be  parabolic  arcs. 

195.  A  BRIEF  GENERAL  ANALYSIS  of  the  method  of  procedure  in 
solving  problems  of  this  kind  is: 

(1)  To  draw  a  general  logarithmic  curve  on  rectangular  coordi- 
nates, the  longest  and  shortest  ordinates  of  which  will  correspond  to 
the  estimated  longest  and  shortest  radii  of  the  cam,  or  the  longest 
and  shortest  radii  of  a  series  of  cams  if  a  series  should  happen  to  be 
under  design. 

(2)  To  compute  the  length  of  rectangular  cam  chart,  as  directed 
in  paragraph  198,  and  to  draw  the  rectangle  on  tracing  cloth  or 
tracing  paper. 

(3)  To   construct   parabolic   arcs   within   the   rectangular   cam 
chart  as  directed  in  paragraph  199. 

(4)  To  place  the  cam  chart  as  now  drawn  on  tne  tracing  cloth, 
over  the  logarithmic  curve,  so  that  the  logarithmic  curve  will  be 
tangent  to  the  two  parabolic  arcs  while  the  bottom  line  of  the  chart  is 
parallel  to  the  abscissa  of  the  logarithmic  curve.     The  distance 
between  the  bottom  of  the  chart  and  the  abscissa  will  be  the  shortest 
radius  of  the  cam. 

196.  The  first  step  in  the  detail  of  the  solution  of  Problem  21  is 
to  construct  a  logarithmic  curve  on  rectangular  coordinates  as  in 
Fig.  127.     This  curve  is  a  perfectly  general  one  and  if  it  is  drawn 
with  a  wide  enough  range  of  ordinates  will  do  for  all  possible  log- 
arithmic-combination cams,  independently  of  all  specific  data.     To 
construct  the  logarithmic  curve  draw  a  horizontal  abscissa  line  0  0', 
Fig.  127,  and  erect  a  series  of  ordinates  one  unit  apart  as  on  both 
sides  of  r,  making  their  length  a   geometrical  progression.      To  do 
this,  make  the  first  ordinate  drawn,  say  0  L,  equal  to  1  unit  and  all 
succeeding  ordinates  such  as  ri,  r%  longer  than  the  preceding  ordinate 
by  using  any  common  multiplier  throughout;    also,  all  preceding 


102 


CAMS 


ordinates  such  as  r',  r",  if  tney  are  necessary,  shorter  by  the  inverse 
of  the  same  ratio.  For  example,  if  0  L  equals  1  and  if  the  common 
multiplier  is  taken  as  1.25  (it  may  be  any  convenient  number),  then 
ri  =  1X1.25  =  1.25,  r2  =  1.25X1.25  =  1.5625.  r3  =  1.5G25  X  1.25  = 


1.953,  etc.;  also/  =  1  X 


1 


1.25 


.8,  r"  =  .8  X     <r  =  .04   etc.      The 


lengths  should  be  accurately  computed  up  to  the  length  of  the 
maximum  radius  of  the  largest  cam  that  is  likely  to  be  used  and  the 
curve  L  G  carefully  drawn. 


FIG.  127.  —  GENEIIAL  LOGARITHMIC  CURVE  SHOWING  SUBTANGENT,  TJsnir 
A  WIDE   RANGE  OF  LOGARITHMIC   CAM  PROBi,r/:»:a 


IN  SOLVING 


197.  The  length  of  the  sub-tangent,  s,  in  Fig.  127,  is  next  found  by 


the  formula,  s  = 


.434 


,  where  m  is  the  common  multiplier  used  in 


log.  m' 
laying  out  the  logarithmic  curve  L  G.    Since  the  value  of  m  is  1.25 

434 

in  this  problem,  .s  =  '-^--  =  4.48.     This  value  of  the  sub-tangent  may 

also  be  found  graphically  by  drawing  tangents  to  the  logarithmic 
curve,  by  eye,  at  several  points  and  taking  an  average  of  the  sub- 
tangents  thus  found.  This  average  value  will  probably  be  close 
enough  for  most  practical  work.  The  tangent  line  at  A  is  shown  at 
A  C,  Fig.  127.  The  length  of  the  sub-tangent,  B  C,  will  be  the  same 
for  each  tangent  line  if  it  is  accurately  drawn. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


103 


198.  A  special  form  of  rectangular  diagram,  Fig.  128,  depending 
on  the  data  is  now  constructed,  its  length  being: 

_  b  TT  s  tan  a 
~I80      ' 

where  I  =  length  of  diagram, 

6  =  assigned  angle  of  action, 

s  =  length  of  sub-tangent  of  the  logarithmic  curve  as  found  in 

the  preceding  paragraph, 
a  =  assigned  pressure  angle. 

p  P       v   s   o 


FIG.  128.  —  RECTANGULAR  CHART  USED  IN  DESIGN  OF  LOGARITHMIC-COMBINATION  CAM 


Taking  the  figures  from  the  data  for   this  problem,   and  the 
value  of  s  as  found  and  substituting  in  the  above  formula, 


1  = 


60  X  3.14  X  4.48  X  .577 
180 


=  2.71. 


The  height  of  the  diagram  is  the  continuous  motion  of  the  fol- 
lower in  one  direction  and  is  1  unit  in  this  problem  as  indicated  at 
R  C,  Fig.  128.  Draw  the  rectangle,  as  shown  at  A  R  C  at  or  near  the 
top  of  a  piece  of  tracing  cloth  or  tracing  paper,  leaving  a  length 
under  it  equal  at  least  to  what  the  short  radius  of  the  cam  is  estimated 
to  be. 

199.  PARABOLIC  EASING-OFF  ARCS  FOR  LOGARITHMIC-COMBINATION 
CAM.  The  length  of  the  rectangular  diagram  is  now  divided  into  at 
least  8  equal  parts  which  are  sufficient  for  practice  problems,  but  in 
practical  applications  at  least  16  divisions  should  be  taken.  A 
diagram  divided  into  8  parts  is  shown  in  Fig.  128.  Construct  a  para- 
bola with  vertex  at  A  and  passing  through  the  midpoint  of  the  dia- 
gram as  at  P.  This  is  done  as  explained  in  detail  in  paragraph  35 


104 


CAMS 


and,  briefly  as  follows :  Divide  A  B  into  a  series  of  equal  parts,  the 
total  number  of  parts  being  equal  to  the  square  of  the  number  of 
construction  spaces  between  A  and  J.  In  this  problem  there  are 
four  construction  spaces  and  so  A  B  is  divided  into  16  equal  parts 
and  the  1st,  4th  and  9th  division  points  are  projected  horizontally 
to  Mj  N  and  0  which  are  points  on  the  parabola.  Construct  the 
similar  parabolic  arc  C  J  in  the  same  way. 

Lay  the  rectangular  diagram  constructed  as  above  on  tracing 
cloth  over  Fig.  127  and  manipulate  it,  always  with  the  line  A  R, 
Fig.  128,  parallel  with  the  line  0  0',  Fig.  127,  until  the  logarithmic 
curve  L  G,  showing  through  the  tracing  cloth,  is  tangent  to  the  two- 
parabolic  arcs.  This  occurs,  in  this  problem,  when  A  R  is  1.55  units 
above  0  0',  and  1.55,  therefore,  is  the  shortest  radius  of  the  pitch 
surface  of  the  cam.  For  precision  work  later  on,  mark  the  points  Y 
and  Z,  Fig.  128,  where  the  logarithmic  arc  comes  tangent  to  the 
parabolic  arcs, 


FIG.  128. — (Duplicate)  RECTANGULAR 
CHART  USED  IN  DESIGN  OF  LOGA- 
RITHMIC-COMBINATION CAM 


FIG.  129. — PROBLEM  21.  LOGARITHMIC- 
COMBINATION  CAM  WITH  PARABOLIC 
ARCS  AT  ENDS 


200.  The  cam  may  now  be  constructed,  drawing  first  the  circle, 
Fig.  129,  having  a  radius  Q  A  of  1.55  units.  Lay  out  the  angle  A  Q  C 
equal  to  the  assigned  60°  and  divide  it  into  equal  spaces  by  as  many 
radial  lines  as  there  are  ordinates  in  Fig.  128.  Transfer  the  ordinates 
L  M,  H  F,  etc.,  from  Fig.  128  to  Fig.  129  and  draw  the  pitch  surface 
of  the  cam  through  the  points  A,  M,  F,  etc.  The  working  surface 
would  be  a  parallel  curve  distant  from  the  pitch  surface  by  the  radius 
of  the  follower  roller.  With  this  cam  there  would  be  uniform  accel- 
eration of  the  follower  from  A  to  Y  where  the  pressure  angle  reaches 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


105 


30°.    This  angle  remains  constant  until  2  comes  into  action,  when 
the  follower  is  uniformly  retarded  to  zero  at  C. 

201.  If  it  is  desired  to  know  the  pitch  circle  of  the  cam  it  may  be 
found  by  noting,  in  Fig.  128,  where  the  logarithmic  arc  comes  tan- 
gent to  the  starting  parabolic  arc.  This  is  at  Y  and  in  this  problem 
it  is  .06  unit  from  the  bottom  of  the  diagram.  This  distance  is  laid 
off  at  A  S  in  Fig.  129  to  obtain  the  pitch  circle  S  T.  If  it  is  desired 
further,  to  obtain  the  cam  chart  which  is  necessary  to  draw  the  veloc- 
ity and  acceleration  diagrams,  it  may  be  found  as  represented  in 
Fig.  74  where  the  length  D  F'  is  equal  to  the  length  of  the  arc  S  T  in 
Fig.  129  when  both  are  drawn  to  the  same  scale.  D  Ff  is  the  pitch 
line  of  the  chart,  and  A  R  is  .06  unit  below  it,  this  value  being  taken 
from  Fig,  128,  The  length  of  the  ordinates,  L  Af ,  H  F,  etc.,  in  Fig.  74 


\ 

*s 

> 

Pitch 

Line 

z 

t 

F'      } 

^*^ 

I/     H      I      J  R 

FIG.  74. — (Enlarged)   LOGARITHMIC-COMBINATION  BASE  CURVE 

are  equal  to  those  in  Fig.  128  when  both  figures  are  drawn  to  the  same 
scale.  It  will  be  noted  that  no  factor  is  given  in  connection  with 
the  cam  chart  for  the  logarithmic  cam  as  it  is  for  other  cams.  There 
is  no  constant  factor;  it  varies  with  each  problem. 

202.  The  rates  of  acceleration  and  retardation  that  will  be  given 
by  the  cam  at  the  ends  of  the  stroke  are  arbitrarily  determined  in 
Fig.  128  by  causing  the  parabolic  arcs  to  pass  through  P  and  J. 
With  the  parabolic  arcs  so  taken  good  average  results  will  be  ob- 
tained, as  compared  with  other  small  cams.     If  different  accelera- 
tions and  retardations  are  desired  for  the  follower  the  point  P  may 
be  located  further  up,  or  further  down,  and  the  cam  will  be  either 
smaller  or  larger. 

203.  EXERCISE  PROBLEM  21a.     REQUIRED  A  LOGARITHMIC-COM- 
BINATION CAM  with  parabolic  easing-off  arcs  which  will  cause  a  fol- 
lower : 

(a)  To  rise  3  units  in  90°  turn  of  the  cam. 

(b)  To  remain  stationary  for  180°  turn  of  the  cam. 


106 


CAMS 


(c)  To  fall  3  units  in  90°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  35°. 

204.  THE  CHARACTERISTICS  OF  A  CAM  HAVING  A  STRAIGHT  BASE 
LINE  have  already  been  considered  in  the  early  part  of  this  book,  in 
paragraph  32.  A  sharp  or  V-edge  sliding  follower  is  the  only  kind 
that  can  be  used  with  the  straight  base  line  for  true  results;  a  roller 
cannot  be  used  for  reasons  explained  in  paragraph  59.  The  form  of 
the  pitch  surface  of  the  cam  that  is  derived  from  the  straight  base  line 
is  the  Archimedean  spiral.  The  straight  base  line  gives  the  smallest 
simple  cam  for  a  given  maximum  pressure  angle.  Its  method 


FIG.  78. — (Enlarged)  STRAIGHT  BASE  LINE 


FIG.  79. — (Enlarged)  PROBLEM  22. 
STRAIGHT  BASE  LINE  CAM 


of  construction  is  illustrated  in  Figs.  78  and  79  for  a  problem  of  the 
following  data: 

205.  PROBLEM  22.     REQUIRED  A   CAM  WITH  A   STRAIGHT-LINE 
BASE  in  which  the  follower : 

(a)  Rises  1  unit  in  60°  turn  of  the  cam. 

(b)  Falls   1    "    "  60°    "     "    "      " 

(c)  Remains  stationary  for  240°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  30°. 

206.  In  accordance  with  formula  (1),  paragraph  29,  the  radius 

i    vx  i  yo 

of  the  pitch  circle  will  be  57.3 — : —  =  1.65  which  is  drawn  at  0  D 

60 

in  Fig.  79.  The  given  angle  of  60°  for  the  rise  is  laid  off  at  D  0  C 
and  divided  into  any  convenient  number  of  construction  parts,  six 
being  shown  by  the  radial  extension  lines  in  the  Figure.  The  first 
line  is  £  of  F  C,  the  second  f  of  F  C,  etc.  Inasmuch  as  no  roller 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


107 


can  be  used  with  this  cam  the  pitch  and  working  surfaces  coincide, 
and  a  V-edge  follower  must  be  used  for  true  results.  The  max- 
imum pressure  angle  occurs  at  the  start  and  grows  smaller  towards 
the  end  of  the  stroke;  in  this  problem  it  diminishes  to  16°  as  indicated 
in  the  Figure. 

207.  EXERCISE  PROBLEM  22a.  REQUIRED  A  CAM  WITH  A  STRAIGHT- 
LINE  BASE  in  which  the  follower : 

(a)  Rises  3  units  in  ]  20°  turn  of  the  cam. 

(b)  Falls  3     "     "  120°     "     "    "     " 

(c)  Remains  stationary  for  120°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  30°. 

208.  THE   STRAIGHT-LINE   COMBINATION   BASE   CURVE,   Fig.   82, 
gives  increasing  velocity  and  acceleration  at  the  beginning  of  the 


Pitch  Line 


-2.27 


Fio.  82   (Enlarged)  STRAIGHT-LINE  COMBINATION  BASE  CURVE 

stroke,  uniform  velocity  and  zero  acceleration  during  a  large  middle 
portion  of  the  stroke,  and  decreasing  velocity  and  retardation  at  the 
end.  The  length  of  the  period  for  uniform  velocity  and  the  amounts 
of  acceleration  and  retardation  depend  entirely  on  the  length  of  the 
easing-off  radius.  This  may  be  taken  at 
any  value.  The  acceleration  diagram  in 
Fig.  85  is  based  on  a  radius  equal  to  the 
follower  motion  as  shown  at  B  A,  Fig.  82. 
The  shorter  this  radius  is  taken,  the  nearer 
the  straight-line  combination  curve  ap- 
proaches the  cam  having  a  straight  base 
line,  Fig.  78,  and  the  action  at  the  beginning 
and  at  the  end  of  the  stroke  becomes  more 
violent.  The  longer  the  easing  off  radius  is 
taken,  the  nearer  the  combination  curve  approaches  the  circular  base 


FIG.  85. — (Duplicate)  ACCEL- 
ERATION DIAGRAM  FOR 
STRAIGHT  -  LINE  -  COMBINA- 
TION CAM 


108 


CAMS 


curve  of  Fig.  98  and  the  smoother  the  action  will  be,  but  in  this  case 
the  cam  will  be  relatively  large.  The  combination  curve  cannot 
be  laid  out  directly  on  the  cam  itself;  the  chart  must  be  constructed 
first  and  the  ordinates  transferred  to  the  cam  drawing.  The  con- 
struction of  a  cam  from  the  combination  curve  is  illustrated  in 
Problem  10,  page  55. 

209.  THE  CRANK  CURVE  BASE,  Fig.  86,  described  in  paragraph  34, 
gives  increasing  variable  velocity  during  the  first  half  of  the  stroke 

and  decreasing  variable  velocity  during 
the  last  half.  The  acceleration  and  re- 
tardation are  also  variable,  being  greatest 
at  the  ends  as  may  be  noted  by  an  in- 

spection of   Fig.    89.     The   suddenness   of 
° 


FIG.    89.  —  (Duplicate) 
CELEBATION  DiAGKA 
CUBVE  CAM 


Ac- 


. 
FOB  the  starting  action  compares  with  that  of 

a  kody  starting  to  fall  under  the  action  of 
gravity,  approximately  as  1.23  is  to  1.00. 

210.  The  crank  curve  is  sometimes  called  THE  HARMONIC  CURVE 
due  to  the  fact  that  it  gives  to  the  follower  a  motion  similar  to  that 
described  by  the  foot  of  a  perpendicular  let  fall  on  the  diameter  of  a 
crank  circle  from  a  crank  pin  moving  with  uniform  velocity  in  that 
circle;    or,  in  other  words,  a  motion  similar  to  that  of  a  crosshead 
which  is  operated  from  a  uniformly  rotating  crank  with  a  T-headed 
or  "  infinite  "  connecting  rod.     It  will  also  be  observed  that  the 
crank  curve  is  a  projection  of  a  helix  onto  a  plane  surface  parallel  to 
the  axis  of  the  helix,  and  is,  further,  a  sine  curve,  or  sinusoid,  in  which 
the  length  or  pitch  is  not  necessarily  equal  to  the  circumference  of 
the  construction  circle. 

211.  EFFECT  OF  CRANK  CURVE  FOLLOWING  ITS  TANGENT  LINE 
CLOSELY.     The  crank  curve  has  the  marked  characteristic,  under 
ordinary  conditions,    of  following   its  tan- 

gent so  closely,  as,  for   example,   on   each 

side  of   E,   Fig.  86,  that  when  the  crank 

curve  chart  is  bent  to  form   the   cam,  as 

explained    in    paragraphs    54    and    55,    a  FlG.  86.—  (Duplicate) 

maximum    pressure    angle   slightly  greater  CHART  CUBVE 

than  30°  is  produced  in  the  cam.     In  the 

case  illustrated  in  Fig.  87  the  pressure  angle  would  still  be  30° 

at  E  but  it  would  be  30°  27'  just  to  the  left  of  E  towards  A  .    If  it  were 

desired  to  keep  the  maximum  pressure  angle  exactly  30°  instead  of 

30°  27',  it  could  be  done  by  moving  all  the  points  from  A  to  C, 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


109 


Fig.  87,  outward  radially  by  the  amount  d  given  in  the  following 
formula: 


FIG.  87. — (Enlarged)  CRANK  CURVE  CAM 

where  d  =  distance  the  points  on  the  pitch  surface,  as  obtained  in 
the  ordinary  way,  would  have  to  be  moved  out  radially 
to  obtain  exact  size  of  crank  curve  cam  for  a  given  max- 
imum pressure  angle. 

h  =  total  rise  of  follower. 

b  =  angle  turned  by  cam  during  the  follower's  total  rise,  in 
radians.  If  b  is  taken  in  degrees  the  number  180  must 
be  used  in  place  of  TT. 

a  =  pressure  angle  in  degrees. 

The  maximum  pressure  angle  of  30°  would  then  occur  where 
the  enlarged  pitch  surface  crosses  the  pitch  circle  which  would  be 
slightly  to  the  left  of  E,  Fig.  87.  The  cam  would  be  .09  larger  in 
maximum  radius,  or  3.19  units  from  0  to  C  instead  of  3.10  as  shown 
and  as  used  in  practice. 

212.  Another  way  of  obtaining  exact  results  with  the  crank  curve 
would  be  to  compute  the  length  of  the  chart  from  the  following 
formula: 


I  =  .5  b  h 


7^  cot 2  a. 


110 


CAMS 


For  the  case  in  hand  I  would  equal  2.77,  which  it  will  be  noted  is  .05 
larger  than  the  practical  value  used  in  Fig.  86.  With  this  length  of 
chart  the  crank  curve  base  line  would  not  reach  a  30°  angle  in  the 
chart  but  the  cam  pitch  surface  would,  at  a  point  just  inside  of  the 
pitch  circle. 

213.  PARABOLA.     This  chart  curve,  Fig.  90,  already  discussed  in 
paragraphs  35  and  36,  gives  uniformly  increasing  velocity  to  the 


16 


II 


rux 


3.46 


F  * 


FIG.  90. — (Enlarged)  PARABOLA  BASE  CURVE 

follower  up  to  mid  stroke  when  the  velocity  is  twice  that  produced 
by  the  straight  base  line  as  illustrated  in  Figs.  92  and  80,  respectively. 
The  follower  has  uniformly  decreasing  velocity  during  the  second 
half  of  its  motion.  Both  the  acceleration  and  the  retardation  are 
uniform  throughout  the  entire  stroke  as  shown  by  the  horizontal 
lines  BD  and  FH  in  Fig.  93. 


Etl 


a- 


FIG.  80.'  FiG.  92. 

FIG.  80. — (Duplicate)  VELOCITY  DIAGRAM  FOR  STRAIGHT  BASE  LINE  CAM 

FIG.  92. — (Duplicate)  VELOCITY  DIAGRAM  FOR  PARABOLA  CAM 
FIG.  93. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  PARABOLA  CAM 

214.  PERFECT  CAM  ACTION.  The  parabola  is  the  only  base  curve 
that  gives  a  theoretically  perfect  motion  so  far  as  inherent  smooth- 
ness of  action  is  concerned.  It  gives  to  the  follower  the  same  gentle 
motion  on  starting  as  a  falling  body  has  when  starting  from  rest,  and 
it  brings  the  follower  to  rest  at  the  end  of  its  stroke  with  the  same 
gentle  action  reversed.  For  this  reason  the  curve  is  sometimes  called 
the  "  Gravity  Curve."  The  curve  for  the  parabola  cam  is  also 
referred  to  by  some  as  the  curve  of  squares  from  the  fact  that  one  set 
of  ordinates  of  the  curve  vary  as  the  square  of  the  time,  as  may  be 
noted  from  the  fact  that  the  construction  numbers  1,  4,  9,  and  16  in 
Fig.  90  are  the  squares  of  1,  2,  3,  and  4,  respectively.  In  Fig.  106 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     111 

which  will  be  described  later,  a  curve  is  used  in  which  the  ordinates 
of  the  curve  vary  as  the  cube  of  the  time. 

215.  THE  PARABOLA  BASE  CURVE  will  also  operate  a  follower 
with  the  least  amount  of  effort  of  any  of  the  base  curves,  due  to  the 
fact  that  the  acceleration  is  constant.     Since  the  mass  is  also  constant 
in  cases  under  comparison,  the  force  required  to  move  the  follower 
will  be  constant  and  may  be  represented  by  1.0  as  shown  in  Fig.  93 
in  comparison  with  a  maximum  of  1.8  for  the  logarithmic-combina- 
tion cam,  Fig.  77;  2.0  for  the  straight-line  combination  curve,  Fig.  85; 
1.2  for  the  crank  curve,  Fig.  89;    1.6  for  the  tangential  curve,  Case  I, 
Fig.  97;  1.5  for  the  circular  curve,  Case  I,  Fig.  101;  and  1.7  for  the 
elliptical  curve,  Fig.  105.     These  figures  are  for  symmetrical  chart 
curves.     Among  the  unsymmetrical  chart  curves  shown  in  Figs.  110, 
114,  etc.,  much  larger  direct  forces  even  may  be  required  to  operate 
the  cam  as  illustrated  by  the  relative  maximum  values  of  2.9  for  the 
circular  curve,  Case  II,  Fig.  113;  4.8  for  the  cube  curve,  Fig.  117; 
and  6.4  for  the  tangential  cam,  Case  II,  Fig.  121. 

216.  COMPARISON  OF  PARABOLIC  AND  CRANK  BASE  CURVES.     While 
the  parabola  base  curve  combines  the  two  highest  theoretical  con- 
siderations, namely  smoothest  possible  motion  and  least  power  for 
operation,  it  has  not  become  so  widely  used  as  the  crank  curve. 
This  may  be  due  to  the  experience  of  builders  of  cams  who  have 
found  that  the  crank  curve  permits  of  a  smaller  cam  for  a  given 
pressure  angle  than  does  the  parabola;  or  for  the  same  size  cams 
the  pressure  angle  is  the  smaller  for  the  crank  curve  and,  therefore, 
does  not  " stick  "  or  "run  hard  "  so  much  as  the  parabola  cam  of 
equal  size.  Figures  on  which  the  above  state- 
ments are  based  may  be  seen  in  Fig.  87  where 

it  is  shown  that  a  maximum  radius  of  3.1 
inches  is  required  for  a  lift  of  1  inch  in  60° 
with  a  maximum  pressure  angle  of  30°  when 
the  crank  curve  is  used;  while  in  Fig.  91  a 
parabola  cam  is  shown  to  require  a  maxi- 
mum radius  of  3.8  inches  for  the  same  data. 
The  crank  curve  has  obtained  some  undue 
comparative  credit  over  the  "parabola"  curve  FlG-  9i.— (Enlarged)  PARABOLA 
on  account  of  the  fact  that  the  "parabola" 

was  constructed  with  spaces  in  some  other  ratio  than  1,  3,  5,  etc. 
While,  for  example,  a  true  parabola  may  be  constructed  with 
spaces  of  1,  2,  3,  instead  of  1,  3,  5,  as  used  in  paragraph  35, 


112 


CAMS 


the  parabolic  curve  of  the  cam  surface  in  the  former  case  will 
not  be  tangent  to  the  circular  part  of  the  cam  surface,  or,  in 
other  words,  the  base  curve  E  A  in  Fig.  90  will  not  be  tangent  to  the 
horizontal  base  line  of  the  chart  at  A  but  will  intersect  it  at  that 
point.  A  "parabola"  cam,  therefore,  with  ordinates  that  are  in 


FIG.  90. — (Duplicate)   PARABOLA  BASE  CURVE 

any  other  ratio  than  1,  4,  9,  etc.,  will  naturally  show  "bright  spots  " 
and  rapid  wear  at  the  beginning  and  end  of  the  parabolic  surface, 
and  this  has  actually  been  erroneously  charged  against  the  true 
practical  parabola  cam. 

217.  A  further  comparison  of  the  parabola  and  crank  base  curves 
shows  that  their  velocity  and  acceleration  lines,  Figs.  88,  89,  92  and 
93,  do  not  differ  in  their  maximum  values  to  such  an  extent,  as  to 


RP 


FIG.  88. — (Duplicate)  VELOCITY  DIAGRAM  FOR  CRANK  CURVE  CAM 
FIG.  89. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CRANK  CURVE  CAM 


FIG.  92. — (Duplicate)  VELOCITY  DIAGRAM  FOR  PARABOLA  CAM 
FIG.  93. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  PARABOLA  CAM 

make  a  noticeable  difference  in  the  action  in  many  cam  applications, 
particularly  where  the  smoothest  motion  is  not  essential  nor  where 
there  is  a  surplus  of  driving  power.  Furthermore,  the  drawing  of  the 
crank  curve  has  appeared  to  some  builders  as  a  much  easier  and 
better-understood  procedure  and  this  has  accounted  some  for  the  use 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     113 

of  the  crank  curve.  It  may  be  observed,  however,  that  the  parabola 
is  really  no  more  difficult  to  draw  than  the  crank  curve,  and  when  it 
is  fully  understood  it  is  quite  certain  that  the  parabola  cam  will  come 
into  a  more  general  use  in  all  cases  except  where  space  is  extremely 
limited,  or  where  special  considerations  of  the  follower  motion  as  to 
spring  or  gravity  action  or  as  to  low  striking  or  seating  velocity,  etc., 
become  especially  desirable.  The  subjects  of  spring  action  and  low 
striking  velocities  will  be  treated  in  paragraph  273,  et  seq. 

218.  TANGENTIAL  BASE  CURVE.     This  base  curve  differs  from  the 
others  in  that  it  cannot  be  readily  used  to  construct  the  cam.     The 
cam  itself  is  drawn  first  by  using  straight  lines  as  the  side  boundaries 
of  the  cam  lobe,  the  straight  lines  being  rounded  off  at  the  ends  by 
arcs  of  circles  or  other  smooth  curves  as  shown  in  Fig.  95.  '"At  the 
inner  ends,  the  straight  lines  are  tangent  to  a  circle  which  has  the 
center  of  rotation  of  the  cam  as  its  center.     The  base  curve  for  this 
cam  is  useful  only  where  it  is  desired  to  find  graphically  the  velocity 
and  acceleration  diagrams,  and  when  it  is  so  used,  it  must  be  derived 
from  the  cam  drawing  as  explained  in  paragraph  225.    The  tangential 
cam  is  perhaps  the  easiest  of  all  cams  to  draw  when  one  is  not  par- 
ticular about  the  maximum  pressure  angle,  but  it  is  apt  to  give  the 
highest  velocities  and  the  greatest  accelerations  of  all  the  cams  when 
it  is  laid  out  "by  eye  "  by  an  inexperienced  person.     To  keep  the 
tangential  cam  under  control  when  being  designed,  requires  either  a 
preliminary  graphical  construction,  or  a  series  of  computations  by 
means  of  formulas  which  will  give  results  that  may  be  laid  out 
directly. 

219.  PROBLEM  23.    TANGENTIAL  CAM,  CASE  I.    Required  a  tan- 
gential cam  in  which  the  follower: 

(a)  Rises  1  unit  in  60°  turn  of  the  cam. 

(b)  Falls  1   "     "  60°     "     "    "     " 

(c)  Remains  at  rest  for  240°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  30°  and  the  end  of  the  lobe 

to  be  rounded  off  by  a  circular  arc. 

Find:  The  shortest  radius  of  pitch  surface  of  cam,  the  length  of 
the  straight-line  portion  of  the  cam  lobe,  the  radius  of  the  rounding 
off  curve  at  the  end,  and  the  largest  size  roller  that  may  be  used. 

220.  THE  GRAPHICAL  METHOD  OF  CONSTRUCTION  FOR  THE  TANGEN- 
TIAL CAM  is  as  follows:  In  a  preliminary  and  separate  drawing,  con- 
struct an  angle  A  0  E,  Fig.  130,  equal  to  the  given  pressure  angle; 


114 


CAMS 


draw  a  line  A  E  at  right  angles  to  0  A  at  any  distance  out,  and  con- 
tinue A  E  until  it  intersects  0  E;  draw  an  angle  A  0  C  equal  to  the 
assigned  angle  of  action;  drop  a  vertical  line  from  E  to  0  C;  draw 
the  arc  E  C  with  L  as  a  center;  draw  the  arc  C  G  with  0  as  a  center, 
and  measure  the  distances  G  A  and  A  0..  Then  G  A  :  h  :  :  A  0  :  s, 
where  h  is  the  assigned  motion  of  the  follower  and  s  is  the  correct 
radius  at  which  to  draw  the  line  A  E  in  the  direct  drawing  of  the  cam. 


FIG.  130.- 


-TANGENTIAL  CAM,  PRELIMINARY  SKETCH  IN  GRAPHICAL  METHOD  OF  CON- 
STRUCTION FOR  DEFINITELY  ASSIGNED  DATA 


In  the  present  illustration  G  A,  Fig.  130,  is  1.33  units  and  A  0  is  4 
units.     Therefore,  in  the  direct  drawing  of  the  cam,  Fig.  95, 

h X AO       1X4 


GA 


1.33 


=  3.00, 


and  this  value  is  laid  off  at  0  A  Fig.  95.  The  pitch  surface  of  the 
cam  A  E  C  is  then  drawn  by  repeating  the  operations  in  precisely  the 
same  order  as  in  the  preliminary  drawing  described  above.  The 
maximum  pressure  angle  will  be  30°  at  E  where  the  circular  easing-off 
arc  is  tangent  to  the  straight  line.  The  maximum  radius  of  the 
roller  would  be  E  L,  but  as  this  would  leave  a  sharp  edge  on  the 
working  surface  of  the  cam,  a  value  of  %  E  L  is  taken  as  the  radius, 
thus  giving  W  N  P  as  the  working  surface  of  the  cam. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


115 


221.  ANALYTICAL  METHOD  OF  CONSTRUCTION  OF  THE  TANGENTIAL 
CAM.  A  direct  drawing  of  the  tangential  cam  may  be  made  from 
value  obtained  from  a  series  of  formulas  having  the  following  nota- 
tion, in  which  all  linear  dimensions  are  in  inches  and  all  angular 


FIG.  95. — (Enlarged)  PROBLEM  23.     TANGENTIAL  BASE  CURVE  CAM,  CASE  1 

dimensions  in  degrees  unless  otherwise  specified.     All  symbols  are 

illustrated  in  Fig.  131  which  is  for  a  general  case: 

h  =  total  motion  of  follower. 

x  =  fraction  of  follower's  motion  while  rolling  on  the  straight  sur- 
face of  the  cam,  or,  fraction  of  stroke  during  which  acceleration 
takes  place. 

a  =  maximum  pressure  angle. 

b  =  time  allotted  by  the  data  to  the  follower  motion,  measured  in 
angular  motion  of  the  cam  in  degrees.  . 

s  =  radius  of  pitch  surface  to  which  the  straight  pitch  line  is  drawn 
tangent. 

t    =  length  of  straight  edge  of  cam  on  both  pitch  and  working  surface. 

p  =  radius  of  pitch  circle. 

d  =  largest  radius  of  pitch  surface  of  cam. 

c  —  angle  turned  through  by  the  cam  when  the  full  motion  of  the 
follower  is  reached,  c  will  equal  b  when  the  straight  part  of 
the  cam  is  not  assigned  in  the  data. 


116 


CAMS 


e   =  radius  of  circular  arc  for  rounding-off  outer  corner  of  pitch  cam. 
r   =  radius  of  roller. 

w  =  radius  of  working  surface  to  which  the  straight  working  line  of 
the  cam  is  drawn  tangent. 


Fia.  131. — TANGENTIAL  CAM,  SHOWING  TERMS  USED  IN  THE  DIRECT  CONSTRUCTION  BY 
THE  ANALYTICAL  METHOD 

222.  When  the  length  of  the  straight  part  of  the  cam  is  not 
assigned  in  the  data,  c  and  b  will  be  equal.  When  the  length  of  the 
straight  part  is  assigned  c  will  figure  out  differently  from  6;  if  it 
comes  less  the  problem  is  possible  with  the  assigned  data;  if  more, 
the  length  of  the  straight  part  must  be  reduced. 

The  general  formulas  are : 


s  = 


P  = 


xh 


sec  a  —  1 
t 


sin  a 


cot  c  =  2 
r  ^  e 


(3) 

(5) 
(7) 


t  =  s  tan  a 


d  =  s  +  h 


e  =  d--r 


sin  c 


w  =  s—r 


(2) 
(4) 

(6) 
(8) 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


117 


223.  With  the  data  of  the  present  problem,  equation  (5)  must 
be  solved  first,  for  it  is  the  only  one  in  which  all  the  terms  but 
one  are  known.  This  formula  is  solved  for  t.  With  t  known,  formula 
(2)  may  be  solved  for  s,  then  formula  (1)  for  x,  and  so  on  in  order 
with  equations  (3),  (4),  (6),  (7),  and  (8).  These  formulas  give  the 
following  values  in  the  present  problem: 


t  =1.73 


x  =     .46 
r  ^  1.5 


p  =  3.46 
w  =  1.5 


224.  With  the  above  values,  the  cam  in  Fig.  95  is  laid  out  in  the 
following  manner:   Lay  off  given  angle  of  60°  at  D  0  C,  draw  circle 


FIG.  95. — (Duplicate)  PROBLEM  23.     TANGENTIAL  BASE  CURVE  CAM,  CASE  1 

having  radius  0  A  equal  to  s,  draw  straight  part  of  cam  A  E  equal  to  t, 
draw  circular  arc  E  C  with  center  on  0  C  and  with  radius  of  L  C 
equal  to  e,  call  r  =  .75e  and  make  A  W  equal  to  it.  Then  W  N  P  is 
the  working  surface  of  the  cam  where  A  W  is  the  radius  of  the  roller. 
The  length  W  N  of  the  straight  part  of  the  working  surface  is  the 
same  as  the  length  of  the  straight  part  of  the  pitch  surface,  and  the 
circular  arc  N  P  of  the  working  surface  has  the  same  center  as  the 
arc  E  C  of  the  pitch  surface.  The  values  d  and  w  are  automatically 
included  in  the  process  of  the  above  described  layout. 


« 


118  CAMS 

225.  If  it  is  desired  to  construct  the  cam  chart,  Fig.  94,  for  the 
tangential  cam  in  order  to  find  the  velocity  and  acceleration  diagrams, 
the  pitch  circle  of  the  cam,  Fig.  95,  should  be  drawn  with  the  radius 
equal  to  0  E  as  computed  above,  and  radial  intercepts  should  be 
placed  at  regular  distances  as  shown  at  H,  /,  etc.,  in  Fig.  95.     Then 

draw  part  of  the  cam  chart  with 
length    equal   to    pitch    arc  D  F, 
when  both  are  to  same  scale,  and 
with    height    equal   to   h.      Draw 
FIG.  94.— (Duplicate)  TANGENTIAL       pitch   line   D  F    on    the    chart    at 
BASE  CURVE,  CASE  i  a    distance    above    A  R    equal    to 

D  A   on  the  cam  when  both   are 

to  the  same  scale.  In  general  the  pitch  line  on  the  chart  will 
not  be  half  way  up,  although  it  appears  so  in  this  problem.  Take 
the  lengths  of  the  radial  lines  at  H,  /,  etc.,  which  are  shown  on 
the  cam  in  Fig.  95  and  lay  them  off  at  equally  spaced  distances  on 
the  chart,  Fig.  94,  and  draw  the  chart  base  curve  A  E  C  through  the 
extremities  of  these  lines. 

226.  THE  TANGENTIAL  CAM  FOR  THIS  CASE  HAS  A  CHARACTERISTIC 
RETARDATION  CURVE  in  that  it  is  convex  downward  as  shown  from  F 
to  H  in  Fig.  97,  while  the  retardation  curves  for  all  other  cams  that 
have  intermediate  maximum  ordinates  are  either  straight  or  con- 
cave.    This  characteristic  may  be  an  advantage  in  some  cam  appli- 
cations and  will  be  referred  to  in  paragraph  273  et  seq.  on  the  use  of 
springs  for  returning  the  follower.     The  pressure  angle  factors  for 
this  curve,  for  the  data  given  in  this  problem,  are:  5.28  for  20°,  3.62 
for  30°,  2.82  for  40°,  2.36  for  50°,  and  2.09  for  60°.     These  factors 
are  used  for  the  ordinates  of  curve  No.  9  in  Fig.  132  which  shows  that 
the  tangential  cam,  for  the  data  of  Problem  23,  has  the  advantage 
of  smaller  size  over  the  parabola,  circular,  elliptical  and  cube  cams 
when  the  lower  range  of  pressure  angles  are  used,  but  that  it  begins 
rapidly  to  lose  this  advantage  at  angles  of  about  36°. 

227.  FURTHER  CHARACTERISTICS  OF  THIS  TANGENTIAL  CAM  that 
may  be  used  to  advantage  in  assigning  data,  are  that  if  the  angle 
turned  through  by  the  cam  is  twice  the  pressure  angle,  the  maximum 
retardation  for  the  circular  easing-off  arc  of  the  cam  will  occur  at 
the  end  of  the  stroke  as  shown  at  C  H,  Fig.  97;  and  that  the  retarda- 
tion at  the  point  on  the  cam  where  the  arc  joins  the  straight  line  will 
be,  .866  C  H  as  shown  at  E  F,  Fig.  97.     If  the  angle  turned  through 
by  the  cam  during  the  motion  of  the  follower  is  greater  than  twice 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


110 


the  pressure  angle  the  retardation  value  will  still  be  a  maximum  at 
the  end  but  will  be  less  than  .866  of  this  value  at  the  point  where 
retardation  begins,  that  is,  E  F  will  be  still  shorter  in  comparison  with 
CH  than  it  is  shown  in  Fig.  97.  This 
condition  has  the  practical  value  in  that  it 
allows  a  lighter-weight,  or  smaller  spring 
to  return  the  follower  where  a  spring  is 
used.  If  the  angle  turned  through  by  the 
cam  during  the  motion  of  the  follower  is 

i,v  ,1  i       .1  FIG.  97.  —  (Duplicate)  ACCEL- 

less  than  twice  the  pressure  angle  the  re-     ERATION  DIAGRAM  FOK  TAN- 

tardation  at  E  F  will  be  greater  than  .866     GENTIAL  CAM 

C  H,  and  if  it  is  much  less  the  retardation 

value  will  be  a  maximum  at  the  point  where  the  easing-off  arc  joins 

the  straight  line,  that  is,  E  F  will  be  greater  than  C  H. 

228.  EXERCISE  PROBLEM  23a.     TANGENTIAL  CAM,  CASE  I.     Re- 
quired a  tangential  cam  in  which  the  follower  : 


(a)  Rises  1%  units  in  50°  turn  of  the  cam. 

(b)  Falls  1J^  units  ''50°     "     "    "     " 

(c)  Remains  at  rest  for  260°  turn  of  the  cam. 

(d)  The  maximum  pressure  angle  to  be  30°,  and  the  end  of  cam 
lobe  eased  off  by  a  circular  arc. 

229.  CIRCULAR   BASE   CURVE,  CASE  I.     This  curve,  Fig.  98,  is 
made  up  simply  of  two  equal  circular  arcs  as  shown  at  A  E  and  E  C. 


s          s 

Fio.  98. — (Enlarged)  CIRCULAR  BASE  CURVE,  CASE  1 

It  is  the  limiting  case  of  the  straight-line  combination  curve  in 
which  the  two  easing-off  arcs  are  so  large  as  to  meet  and  eliminate 
the  intermediate  straight  line  entirely.  The  circular  base  curve 


120 


CAMS 


gives  variable  velocity  and  acceleration  to  the  follower  the  first  half 
of  the  follower  stroke,  and  also  variable  velocity  and  retardation 
during  the  last  half,  as  shown  in  Figs.  100  and  101.  It  will  be  noted 


Fio.   100  FIG.   101 

FIG.  100. — (Duplicate)  VELOCITY  DIAGRAM  FOB  CIRCULAR  BASE  CURVE  CAM 
FIG.  101. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CIRCULAR  BASE  CURVE  CAM 

that  the  circular  curve,  and  the  elliptical  curve  shown  in  Fig.  102, 
give  nearly  the  same  sized  cams  and  that  the  velocity  and  acceleration 
diagrams  for  each  are  quite  similar.  With  the  circular  base  curve, 
the  radial  distances  on  the  cam  at  D,  H,  /,  J,  Fig.  99,  cannot  be  found 


FIG.  99. — (Enlarged)  PROBLEM  24.     CIRCULAR  BASE  CURVE  CAM,  CASE  1 

directly  except  by  means  of  the  chart  or  by  computation.  For 
graphical  construction  it  is  necessary  to  draw  the  chart,  Fig.  98, 
first  and  it  is  then  a  simple  matter  to  transfer  the  ordinates  at  H,  I,  J, 
to  Fig.  99.  The  length  of  the  chart  for  a  maximum  pressure  angle 
of  30°  is  3.73  times  the  motion  of  the  follower. 

230.  The  length  of  radius  for  the  equal  arcs  in  the  circular  base 
curve  is  3.73  times  the  follower  motion  for  a  30°  maximum  pressure 
angle.  To  find  the  length  of  radius  for  any  other  maximum  pressure 
angle,  use  the  formula : 


2(1  -  cos  a)' 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 

where  r  =  the  desired  radius, 

a  =  the  desired  maximum  pressure  angle, 
and  h  =  the  given  follower  motion. 

TABLE  FOR  CIRCULAR  BASE  CURVE 


121 


For  Maximum  Pressure 
Angle  of 

Radius  of  Arc  is 

20° 

8.29  A 

30° 

3.73  h 

40° 

2.14/t 

50° 

1.40/i 

60° 

1.00  h 

231.  PROBLEM   24.    REQUIRED   A   CIRCULAR  BASE   CURVE   CAM 
that  will  cause  the  follower  to: 

(a)  Rise  1  unit  in  60°  turn  of  cam. 

(b)  Fall  1     "    "  60°  "     "     " 

(c)  Remain  stationary  for  240°    "     "     " 

(d)  With  a  maximum  pressure  angle  of  30°. 

232.  The  general  description  of  the  circular  base  curve  given  in 
the  two  preceding  paragraphs  will  doubtless  give  all  the  necessary 
information  for  the  solution  of  this  problem  so  that  only  a  brief 
order  of  procedure  will  be  given  here.     The  total  length  of  chart  is 


1  X  3.73  X  -(  r  =  22.38. 

DU 

One-sixth  of  this  length  is  shown  in  Fig.  98.     The  radius  of  the  cir- 
cular arc  A  E,  which  is  the  same  as  E  C,  is 


1 


1 


2(1  -  cos  30°)       2(1  -  .866) 


=  3.73. 


Draw  eight  equally  spaced  ordinates  as  at  H,  I,  J,  etc.,  Fig.  98.     The 
radius  of  the  pitch  circle  of  the  cam  is, 


22,38 


2  X  3.14 


j  =  3.56, 


122 


CAMS 


as  drawn  at  0  D  in  Fig.  99.  Divide  the  assigned  arc  of  action  D  F, 
which  is  60°,  into  eight  equal  parts  as  at  H,  /,  /,  etc.  On  the  radial 
lines  at  each  of  these  points  lay  off  the  corresponding  ordinates  from 
H,  I,  J,  etc.,  in  the  chart,  Fig.  98,  thus  obtaining  the  pitch  surface 
A  EC,  Fig.  99. 

233.  In  some  cases  it  may  happen,  when  the  circular  base  curve 
is  assigned,  that  the  length  and  height  only  of  the  rectangular  chart 
enclosing  the  circular  curve  will  be  known  and  it  may  be  desired  to 
compute  the  radius  and  the  pressure  angle  for  the  circular  arc  that 
must  be  used.  For  example,  in  Fig.  98,  assume  that  A  R  and  R  C 
are  the  only  known  values  and  it  is  desired  to  find  the  proper  radius 
of  the  arc  EC  and  the  pressure  angle  that  will  exist  at  E.  The 
radius  may  be  readily  computed  by  simple  geometry,  for,  the  two 


FIG.  98. — (Duplicate)   CIRCULAR  BASE  CURVE,  CASE  1 

triangles  C  F  E  and  C  T  S  will  be  similar  in  all  cases  and,  therefore, 
SC  :  EC  :  :  TC  :  FC.  Since  E  F  and  F  C  are  equal  to  one- 
half  of  A  R  and  R  C,  respectively,  their  values  are  known  and 
E  C  =  V  E  F2  +  F  C2.  The  length  of  T  C  is  one-half  of  E  C. 
The  radius  of  the  circular  arc  will  be 


SC  = 


EC  X  TC 
FC       ' 


234.  In  order  to  obtain  the  pressure  angle,  for  the  case  given  in 
the  preceding  paragraph,  simple  trigonometry  is  required,  and  in 
using  the  trigonometry,  the  length  of  the  radius  may  also  be  obtained 
even  more  readily  than  by  geometry.  The  method  is  as  follows: 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     123 

In  Fig.  98  the  angles  C  S  T  and  E  S  T  are  each  equal  to  one-half  the 
angle  C  S  E  which  is  the  pressure  angle  and  is  designated  by  a  in 
the  following  formulas.  The  triangles  C  E  F  and  C  S  T  are  similar 
in  all  cases.  Therefore,  a  may  be  found  by  the  following  formula  : 

1  CF 


With  a  known,  the  radius  of  the  arc  E  C  may  also  be  found  as 
follows: 

E  F 

E  S  =  =-?-  =  CS. 
sin  a 

235.  EXERCISE  PROBLEM  24a.     Required  a  circular  base  curve 
cam  which  will  cause  the  follower  to  : 

(a)  Move  out  3  units  in  90°        turn  of  the  cam. 

(b)  Remain  stationary  for  195°     "     il     "     " 

(c)  Move  in  3  units  in  75°  "     ll    "     " 

(d)  With  a  maximum  pressure  angle  of  40°. 

236.  ELLIPTICAL  BASE  CURVE.     The  elliptical  base  curve  gives 
variable  velocity  and  variable  acceleration  to  the  follower.     By  using 
different  ratios  for  the  horizontal  and  vertical  axes  of  the  ellipse  on 
which  the  curve  is  based,  the  velocity  of  the  follower  may  be  made 
to  increase  rapidly  or  slowly  at  the  start,  and  the  cam  may  be  made 
small  or  large  and  still  not  exceed  a  given  maximum  pressure  angle. 

237.  ELLIPTICAL  BASE  CURVE,  RATIO  7  TO  4.     As  stated  in  the  pre- 
ceding paragraph  the  elliptical  cam  may  be  based  on  ellipses  having 
various  proportions  between  then*  major  and  minor  axes.     When  the 
proportions  are  as  7   :  4,  as  in  Fig.  102  where  F  G  =  7  and  F  C  =  4, 
the  length  of  the  chart  will  be  3.95  times  the  travel  of  the  follower  for 
a  maximum  pressure  angle  of  30°.     The  cam  will  be  larger,  but  the 
velocity  of  the  follower  will  be  less  at  starting  and  stopping  and 
greater  at  midstroke  than  for  any  of  the  cams  described  thus  far. 
If  a  still  lower  starting  and  stopping  velocity  is  desired  with  an 
elliptical  cam,  it  may  be  obtained  by  making  the  ratio  of  horizontal 
to  vertical  axes  on  the  chart  as  8  :  4,  9   :  4,  or  greater,  instead  of 
7   :  4  as  here  used.     The  drawbacks  to  increasing  the  ratios  above 
7   :  4  are  increased  size  of  cam  and  high  velocity  at  midstroke  for  a 
given  pressure  angle. 


124 


CAMS 


238.  ELLIPTICAL  BASE  CURVE,  RATIO  2  to  4.  The  cam  produced 
from  the  elliptical  base  curve  is  shown,  in  the  preceding  paragraph, 
to  give  a  certain  characteristic  action  to  the  follower  when  the  ratio 


c  L 


FIG.  102. — (Enlarged)  ELLIPTICAL  BASE  CURVE 

of  the  horizontal  axis  to  the  vertical  axis  is  7  to  4.  When  the  ratio 
is  2  to  4,  a  totally  different  characteristic  follower  action  is  obtained 
as  may  be  determined  by  a  process  of  construction  similar  to  that 
shown  in  Figs.  102  and  103.  The  cam  itself,  with  a  ratio  of  2  to  4, 

will  be  much  smaller  for  a  given 
pressure  angle,  as  may  be  seen  by 
comparing  the  abscissae  of  curves  5 
and  11  in  Fig.  132.  Where  it  is 
desired  to  use  a  very  small  cam  for  a 
given  pressure  angle,  the  2  :  4  ellip- 
tical curve  will  have  an  advantage 
over  the  ordinary  straight-line  com- 
bination curve  above  27°  as  may  be 
noted  from  an  inspection  of  curves 
5  and  6,  Fig.  132;  but  it  is  at  a 
disadvantage  compared  with  the  log- 
arithmic-combination cam  at  all  pressures  angles  as  is  shown  by  a 
comparison  of  curves  2  and  5. 

239.  ELLIPTICAL  BASE  CURVE  MAY  BE  MADE  EQUIVALENT  TO 
NEARLY  ALL  OTHER  BASE  CURVES.  Since  the  elliptical  base  curve 
may  be  constructed  with  any  ratio  of  horizontal  to  vertical  axes,  it 
has  a  range  of  usefulness  over  the  entire  field  covered  by  all  the  other 
base  curves  except  the  logarithmic  curve.  When  the  horizontal  axis 
of  the  ellipse  is  zero,  the  elliptical  base  curve  coincides  exactly  with 
the  straight-line  base.  As  the  horizontal  axis  increases  in  length, 
the  vertical  axis  remaining  constant,  the  elliptical  base  curve  crosses 
the  straight-line  combination  curve.  When  the  horizontal  axis  of 
the  ellipse  equals  the  vertical  axis,  the  elliptical  base  curve  is  identical 
with  the  crank  curve.  As  the  horizontal  axis  continues  to  increase, 
the  elliptical  curve  approximates  very  closely  indeed  to  the  parabola 


FIG.  103. — (Enlarged)  ELLIPTICAL 
BASE  CURVE  CAM 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


125 


when  the  ratio  of  horizontal  to  vertical  axes  is  as  11  to  8.  A  further 
general  characteristic  of  the  elliptical  curve  is  that  the  starting  and 
stopping  velocities  grow  smaller,  and  also  the  accelerations  or  start- 
ing and  stopping  forces  grow  smaller  as  the  horizontal  axis  of  the 
ellipse  grows  larger. 

240.  CUBE   BASE   CURVE,   SYMMETRICALLY  APPLIED.    The  cube 
base  curve,  Fig.   106,  is  similar  in  method  of  construction  to  the 


FIG.  106. — (Enlarged)  CUBE  BASE  CUBVE,  CASE  1 

parabola  base  curve,  the  only  difference  being  that  the  cubes  of  the 
numbers  1,  2,  3,  etc.,  instead  of  the  squares,  are  used  as  ordinates  of 
the  curve.  The  cube  curve  gives  extremely  low  and  slowly  increasing 
motion  to  the  follower  at  the  start  as  may  be  noted  by  an  inspection 
of  the  velocity  curve  A  E,  Fig.  108,  which  shows  the  distinguishing 
characteristic  that  the  velocity  curve  is  tangent  to  the  base  line. 
The  cube  curve  is  the  only  one  that  gives  uniformly  increasing  accel- 
eration to  the  follower,  starting  from  zero,  as  indicated  by  the  straight 


FIG.  109. 
Eld.  108. 

FIG.  108. — (Duplicate)  VELOCITY  DIAGRAM  FOR  CUBE  CAM 
FIG.  109. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CUBE  CAM 

inclined  line  A  D  in  Fig.  109.  The  disadvantage  of  the  cube  curve, 
however,  is  that  it  gives  an  extremely  large  cam  for  a  given  maximum 
pressure  angle,  if  it  is  used  in  the  same  way  that  the  preceding  curves 
are  used,  that  is,  if  it  is  made  up  of  two  similar  arcs  placed  in  reverse 
order.  If  the  cube  curve  were  so  drawn  it  would  be  made  up  of 
two  arcs  similar  to  A  E,  Fig.  106,  and  the  pressure  angle  factor  would 
be  5.20  as  compared,  for  example,  with  3.46  for  the  parabola,  and  the 
maximum  radius  of  the  cam  would  be  5.47  against  3.80  for  the  para- 
bola. Because  of  the  similarity  of  method  of  construction  of  the 


126  CAMS 

cube  curve  and  the  parabola,  and  because  the  large  size  of  the  sym- 
metrical cube  cam  renders  it  impractical  for  most  cases,  its  drawing 
will  be  omitted,  and  instead,  a  modified  and  more  practical  con- 
struction of  the  cube  cam  will  be  illustrated  and  explained  in  the 
following  paragraphs. 

241.  CUBE  BASE  CURVE  UNSYMMETRICALLY  APPLIED  FOR  BEST 
ADVANTAGE.     This  modified  cube  curve  will  be  referred  to  as  CUBE 
CURVE,  CASE  I.     Its  features  are  that  it  retains  the  very  low  starting 
values  of  the  regulation  or  symmetrical  cube  cam,  and  at  the  same 
time  keeps  down  the  size  of  the  cam  by  using  che  regulation  cube 
curve  for  the  first  half  of  the  follower's  motion  and  then  using  a 
short  arc  of  another  cube  curve  for  the  retardation  in  such  a  way  that 
the  maximum  acceleration  and  retardation  values  shall  be  equal. 
In  order  to  use  this  base  curve  several  formulas  are  necessary  and 
they,  together  with  their  notation,  are  given  in  the  following  par- 
agraph. 

242.  NOTATION  AND  FORMULAS  FOR  CUBE  CURVE  CAM,  CASE  I: 
h  =  distance  moved  by  the  follower. 

a  =  pressure  angle. 

I  =  length  of  part  of  cam  chart  corresponding  to  follower's  motion. 
x  =  length  of  cam  chart  during  which  acceleration  takes  place. 
xi,X2  .  .  .  =  arbitrary  lengths  of  cam  chart  taken  for  purposes  of 

constructing  chart  base  curve. 
2/i,  2/2  .  .  •  =  length  of  ordinates  of  cam  chart  corresponding  to  the 

values  of  £1,  £2  .•  •  •  • 
r  =  radius  of  pitch  circle  of  cam. 

b  =  angle  turned  through  by  cam  in  degrees  during  follower's  mo- 
tion. 

The  general  formulas  are: 
I  =  2.427  h  cot  a     ...     (1)        x=  .618 1 (2) 

\3 
I.  /*" 

h  [-, 


y  =  — y=r- —  from  zero  to  x (3) 

2  v  o       4 


from  x  to  I (4) 


3  —  v  5 


180  1  ,-. 

r  =  —  r-  ............    .......     (5) 

7TO 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


127 


243.  PROBLEM  25.  CUBE  CURVE  CAM,  CASE  I.  Required  a  cube 
curve  cam  with  unsymmetrical  cube  curve  arcs  in  which  the  follower 
shaU: 

(a)  Rise  1  unit  in  60°  turn  of  the  cam. 

(b)  Fall  1     "    "  60°     "     "    "     " 

(c)  Remain  stationary  for  240°  turn  of  the  cam,  and 

(d)  The  maximum  pressure  angle  shall  be  30°. 

Substituting  the  values  given  in  the  data  in  the  formulas  in  the 
preceding  paragraph,  I  =  4.20,  x  =  2.60  and  r  =  4.0.  With  these 
values,  the  rectangle  A  B  C  R,  Fig.  106,  for  the  cam  chart  may  be 


FIG.   106. — (Duplicate)  CUBE  BASE  CURVE,  CASE  1 

drawn,  A  R  being  made  equal  tol,AX  equal  to  x,  and  R  C  equal  to  h. 
The  curve  A  E  may  be  drawn  graphically  by  dividing  A  X  into  four 
equal  parts,  A  D  into  four  unequal  parts,  as  shown  in  Fig.  106,  and 
projecting  the  division  points  until  they  meet,  as  at  K.  A  D,  which 
is  one-half  of  A  B,  is  divided  into  the  four  unequal  parts  as  follows: 
Draw  a  straight  line  A  G  in  any  convenient  direction  about  as  shown; 
make  its  length  64  units  according  to  any  convenient  scale;  with  the 
scale  still  in  place  mark  the  1st,  8th  and  27th  division  points  on  A  G 
and  from  each  of  these  points  draw  lines  parallel  to  G  D  until  they 
intersect  the  side  A  D  of  the  rectangle;  from  the  latter  points  draw 
horizontal  lines  until  they  intersect  their  corresponding  ordinates, 
as  at  K.  Or,  the  values  of  these  ordinates,  as  at  J  K,  may  be  com- 
puted by  formula  (3)  of  the  preceding  paragraph  by  substituting  the 
following  values  for  x  :  x\  =  %x,  X2  =  %x,  z3  =  %x.  The  computed 
values  of  yi,  2/2,  2/3,  are  .008,  .063,  .211,  respectively,  and  these  are 
laid  off  at  H,  /,  and  J  in  Fig.  106. 

244.  The  portion  of  the  cube  curve  from  E  to  C,  Fig.  106,  is  found 
by  taking  a  series  of  any  number  of  equally  spaced  ordinates,  four 
being  used  in  this  problem  and  one  of  them  marked  at  T  S.  The 
values  of  these  ordinates  are  computed  from  formula  (4)  of  para- 
graph 242,  and  are  as  follows:  y±  =  .50,  y5  =  .71,  y&  =  .87  (shown 


128 


CAMS 


SitS  T),  and  ?/7  =  -95.     The  corresponding  values  of  x±,  x5.  .  .  which 
were  substituted  for  x  in  equation  (4)  in  obtaining  these  values  were 

£4  =  X,  X5  =  X  +  Y±  (I  —  X),  XQ  =  X  +  J^(Z  —  x) ,  etc. 

245.  The  pitch  circle  of  the  cam  is  drawn  with  0  Z),  Fig.  107,  as  a 
radius  and  is  equal  to  r  =  4.00,  obtained  from  equation  5.  The 
values  as  found  for  the  cam  chart  may  be  now  transferred  to  the  cor- 


L 


FIG.  107. — (Enlarged)  PROBLEM  25.     CUBE  BASE  CAM,  CASE  1 

respondingly  placed  radial  lines  from  A  to  R,  or  the  values  as  com- 
puted from  formulas  (3)  and  (4)  may  be  laid  off  directly  on  these 
radial  lines  without  drawing  the  cam  chart  at  all. 

246.  The  characteristic  velocities,  accelerations  and  retardations 
produced  by  this  case  of  the  cube  curve  cam  are  shown  in  Figs.  108 
and  109,  respectively.  From  the  latter  it  may  be  seen  that  the 


FIG.  109. 
FIG.  108. 

FIG.   108. — (Duplicate)  VELOCITY  DIAGRAM  FOR  CUBE  CAM 
FIG.  109. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CUBE  CAM 

acceleration  and  retardation  lines,  A  D  and  F  H,  respectively,  are 
straight  inclined  lines,  characteristic  of  the  cube  curve,  as  pointed 
out  in  paragraph  240.  When  the  retardation  line  F  H  is  extended, 
as  shown  by  the  long-dash  line,  Fig.  109,  it  passes  through  the  zero 
point  of  the  diagram.  A  cam  with  this  characteristic  may  have 
particular  advantages  in  some  instances,  one  of  which  will  be  referred 
to  later  in  the  discussion  of  the  relative  strength  of  springs  necessary 
to  return  the  follower, 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


129 


247.  EXERCISE  PROBLEM,  25a.     CUBE  CURVE,  CAM,  CASE  I.     Re- 
quired a  cube  curve  cam  in  which  the  follower : 

(a)  Moves  up  1  unit  in  50°  turn  of  the  cam. 

(b)  Movesdownl  "    "    50°     "     "    "      " 

(c)  Remains  stationary  for  260°  turn  of  the  cam,  and  in  which 

(d)  The  maximum  pressure  angle  shall  be  30°. 

248.  CAMS  SPECIALLY  DESIGNED  FOR  LOW-STARTING  VELOCITIES. 
In  cams  where  the  change  in  velocity  of  the  follower  during  the  latter 
part  of  its  travel  may  take  place  rapidly  the  early  motion  of  the  fol- 
lower may  be  made  both  very  low  and  very  gradual.     These  condi- 


FIG.  114. — (Duplicate)   CUBE  BASE  CURVE,  CASE  2 

tions  as  to  velocity  may  be  obtained  by  giving  more  than  half  the 
stroke  to  the  acceleration  of  the  follower,  instead  of  one-half  as  has 
been  the  case  in  all  preceding  problems.  In  Figs.  110  and  114,  are 
illustrated  special  cases  of  the  circular  and  cube  base  curves  in  which 
the  follower  is  permitted  to  accelerate  during  %  of  its  stroke,  while 
its  retardation  takes  place  in  the  last  quarter  of  the  stroke.  In  these 


TIG.   112.  FIG.  116. 

FIG.  112. — (Duplicate)  VELOCITY  DIAGRAM  FOR  CIRCULAR  BASE  CURVE  CAM,  CASE  2 
FIG.  116.— (Duplicate)  VELOCITY  DIAGRAM  FOR  CUBE  CAM,  CASE  2 

two  cases  the  velocities  at  midstroke  are  approximately  1.2  and  1.0, 
respectively,  as  may  be  noted  from  the  dash  line  construction  in 
Figs.  112  and  116,  respectively,  against  2.2  and  1.7  as  shown  for 
similar  basic  curves  in  Figs.  100  and  108. 

249.  PROBLEM  26.     CIRCULAR  BASE  CURVE  CAM,  CASE  II.     Re- 
quired a  cam  with  a  circular  base  curve  in  which  the  follower  shall: 

(a)  Rise  1  unit  in  60°  turn  of  the  cam. 

(b)  Fall   1      "    "  60°    "     "     "     " 


130 


CAMS 


(c)  Remain  stationary  for  240°  turn  of  the  cam. 

(d)  Accelerate  for  %  of  its  stroke,  and  in  which 

(e)  The  maximum  pressure  angle  shall  be  30°. 

250.    FOR  A  GRAPHICAL  METHOD  OF  CONSTRUCTION  OF  CASE  II  OF 

THE  CIRCULAR  BASE  CURVE  CAM,  draw  the  cam  chart  as  in  Fig.  110 


making  its  length  I 


h  X  f  X  360 


,  where 


I  =  total  length  of  chart. 

h  =  height  of  chart. 

/  =  pressure  angle  factor. 

b  =  angle  during  which  follower  motion  takes  place. 

In  this  problem  I  =  1  X  37,3  X  36°  =  22.38. 

t)U 

251.  One-sixth  of  the  chart  is  shown  in  Fig.  110  at  A  R.     Lay  off 
the  height  A  B  equal  to  one  unit  and  mark  the  point  D  so  that 


FIG.  110. — (Enlarged)  CIRCULAR  BASE  CURVE, 
CASE  2 


FIG.  111. — (Enlarged)  PROBLEM 
26.  CIRCULAR  BASE  CURVE 
CAM,  CASE  2 


A  D  =  t,  where  t  equals  fraction  of  stroke  assigned  for  acceleration. 
Draw  D  F.  Mark  the  point  X  on  A  R  so  that  A  X  =  t  X  A  R. 
Draw  X  E.  Through  E  draw  an  inclined  line  making  an  angle  with 
X  E  equal  to  the  assigned  pressure  angle.  Where  this  inclined  line 
meets  the  lines  A  B  and  C  R  will  be  the  centers  for  the  circular  arcs 
making  up  the  base  curve.  These  center  points  will  be  at  Y  and  at  S 
respectively.  Draw  the  circular  arcs  A  E  and  E  C.  Divide  D  E 
into  a  convenient  number  of  equal  parts,  as  at  H,  I  .  .  .  and  draw 
ordinates  to  the  circular  arc  A  E.  Do  the  same  with  E  F. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     131 

Construct  the  pitch  circle  of  the  cam  with  a  radius, 

l  22'38 


as  shown  in  Fig.  111.  Lay  ofiDOF  equal  to  the  assigned  motion 
angle,  which  is  60°  in  this  problem.  The  arc  D  F  will  be  equal  in 
length  to  the  line  D  F  in  the  chart  when  both  are  drawn  to  the  same 
scale.  Make  D  E  on  the  arc  equal  to  D  E  on  the  chart  and  divide 
the  arc  D  E  into  the  same  number  of  equal  parts  as  the  line  D  E. 
Draw  radial  lines  at  the  division  points  H,  /,/,...  and  transfer 
the  ordinates  from  the  chart  to  these  radial  lines,  thus  obtaining  the 
pitch  surface  of  the  cam  from  A  to  E.  Do  likewise  to  obtain  the 
arc  E  C  of  the  cam. 

252.  THE  CIRCULAR  BASE  CURVE,  CASE  II,  GIVES  A  SMALLER  CAM 
than  does  case  I,  although  both  have  the  same  pressure  angle  factor 
and  the  same  chart  length.  The  maximum  radius  of  the  cam  for 


FIG.  99.  —  (Duplicate)  CIRCULAR  BASE  CURVE  CAM,  CASE  1 


case  II  is  3.81  against  4.06  for  case  I  as  shown  in  Figs.  Ill  and  99 
respectively.  The  reduction  in  size  in  case  II  is  due  to  the  fact  that 
the  pitch  line  D  F  on  the  cam  chart  is  higher  up  in  the  present  case, 
and,  consequently,  that  more  of  the  pitch  surface  falls  inside  of  the 
pitch  circle  than  in  Fig.  99.  The  pitch  circle  is  the  same  size  in  both 
cases. 

253.  COMPUTATION  FOR  THE  LENGTHS  OF  THE  RADII  for  the  arcs 
A  E  and  E  C  in  the  cam  chart  in  Fig.  1  10  may  be  made  by  the  fol- 
lowing formulas  if  desired,  instead  of  finding  them  graphically  as 
explained  in  paragraph  251. 


and 


w  C4IJLJLVA  V^      f^f        _«  • 

1  —  cos  a  1  —  cos  a 


132 


CAMS 


where  a  equals  the  assigned  pressure  angle,  h  equals  follower  motion, 
and  t  equals  fraction  of  stroke  assigned  to  acceleration. 

254.  EXERCISE    PROBLEM    26a.     CIRCULAR    BASE    CURVE    CAM, 
CASE  II.     Required  a  cam  with  a  circular  base  curve  in  which  the 
follower  shall: 

(a)  Rise  2  units  in  75°  turn  of  the  cam. 

(b)  Fall  2      "     "  75°     "     "    "      " 

(c)  Remain  stationary  for  210°  turn  of  the  cam. 

(d)  Accelerate  for  .7  of  its  stroke,  and  in  which 

(e)  The  maximum  pressure  angle  shall  be  30°. 

255.  THE  USE  OF  THE  CUBE  CURVE  for  obtaining  extremely  low 
starting  velocities  is  illustrated  in  Fig.  115.     The  cam  is  built  up 


X* 

?   ^        ( 

• 

fr< 

Pitch  Line                                                \ 

^$"*\M 

n  ; 

\    D 

H          I           J                        1     .^ 

t        L 

, 

4 

V^ 

64V 

— 

\*J^?* 

A 

8r 

A 

0   f)(\                                                      •>- 

\ 

*  \ 

\ 

FIG.  114. — (Enlarged)  CUBE  BASE  CURVE,  CASE  2 

n 


I 


VI 


FIG.  115. — (Enlarged)  PROBLEM  27. 
CUBE  BASE  CURVE  CAM,  CASE  2 


FIG.  117. — (Duplicate)  ACCELERATION 
DIAGRAM  FOR  CUBE  CAM,  CASE  2 


from  a  specially  long  arc  of  the  cube  base  curve  and  it  has  a  short 
circular  base  arc  for  easing  off  at  the  end.  The  chart  and  the  base 
curve  for  th^s  cam  are  shown  in  Fig.  114.  The  low-starting  velocities 
are  due  to  the  fact  that  the  follower  has  %  of  its  stroke  to  reach  max- 
imum velocity.  This  gives  only  %  stroke  for  retardation  which  attains 
a  very  high  value  near  the  end  of  the  stroke  ranging  from  4.8  to  3.2, 
as  shown  in  Fig.  117.  This,  of  course,  becomes  the  acceleration 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS     133 

value  at  the  beginning  of  the  return  stroke.  Herein  lies  the  disad- 
vantage of  this  cam.  It  is  useful  only  where  extremely  slow  starting 
velocity  is  required  at  one  end  of  the  stroke  and  where  a  rapid  change 
of  velocity  at  the  other  end  of  the  stroke  is  immaterial.  It  would 
require  a  powerful  spring  to  keep  the  follower  roller  in  contact  with 
the  cam  at  high  speeds,  and  if  it  were  used  on  a  positive  drive  cam 
would  cause  rapid  wear  at  the  beginning  of  the  return  stroke. 

256.  PROBLEM  27.  CUBE  CURVE,  CASE  II.  Required  a  cube 
curve  cam  with  a  circular  arc  for  easing-off  radius  in  which  the  fol- 
lower : 


(a)  Rises  1  unit  in  60°  turn  of  the  cam. 

(b)  Falls  1    "     "  60°     "     "    "      " 

(c)  Remains  stationary  for  240°  turn  of  the  cam. 

(d)  Accelerates  during  %  of  the  stroke. 

(e)  The  maximum  pressure  angle  to  be  30°. 

257.  In  solving  the  above  problem  the  length  A  X,  Fig.  114,  of 
that  part  of  the  chart  which  is  given  over  to  the  cube  curve  is  first 
found  by  the  formula, 

3tt 

xi  =  -      -  where 
tan  a, 

t    =  the  fractional  part  of  the  follower's  motion  devoted  to  accelera- 

tion. 

h    =  the  total  motion  of  the  follower. 
a    =  the  pressure  angle. 

xi  =  the  length  of  chart  under  the  cube  curve. 
X2  =  the  length  of  chart  under  the  circular  easing-off  arc, 
Substituting  the  values  given  in  problem  27, 

3  X  .75  X  1 
Xl  - 


258.  The  length  X  R  of  chart,  Fig.  114,  necessary  for  the  easing 
off  circular  arc  may  be  computed  by  the  formula, 

_  h(l  -  t)        .25 
" 


Or,  the  length  X  R  may  be  found  directly  by  drawing  NEK 
so  that  it  is  tangent  to  the  cube  curve  at  E.    The  angle  KEF  will 


134 


CAMS 


then  be  equal  to  the  pressure  angle.  Make  K  C  equal  to  E  K.  The 
point  C  will  then  be  at  the  end  of  the  chart.  The  center  for  the  arc 
E  C  will  then  be  on  the  line  C  R  extended,  and  at  a  point  S  which 
must  also  be  on  the  perpendicular  to  N  E  K. 

259.  To  find  points  on  the  cube  base  curve  A  E,  Fig.  114,  divide 
D  E  into  any  convenient  number  of  equal  parts,  six  being  used  in  the 
illustration.  Draw  vertical  lines  through  each  of  the  division  points 
as  at  H ,  I,  ...  Draw  a  line  A  G  inclining  upward  from  A  in  any 
convenient  direction  and  make  the  distance  A  G  equal  to  the  cube  of 
the  number  of  construction  parts.  Six  parts  having  been  chosen 
in  this  problem,  A  G  will  be  equal  to  the  cube  of  6,  or  216  units  in 
length  laid  off  to  any  convenient  scale.  At  the  same  time  lay  off  the 
division  points  1,  8,  27,  etc.,  which  are  the  cubes  of  1,  2,  3.  etc.  Draw 


216 


n 

1 
if 

V 

"p    K        C 

Pitch  Line                                                \ 

23? 

F\ 

— 

H 

L 

3      ^^ 

\        L 

A 

A 

*                                                          4.83  

FIG.  114. — (Duplicate)   CUBE  BASE  CURVE,  CASE  2 

the  line  G  D,  and  then  draw  lines  parallel  to  it  through  the  points 
1,  8,  27,  etc.,  until  they  intersect  A  D.     Project  these  intersecting 
points  horizontally  until  they  meet  the  corresponding  verticals  from 
#,/,...,  thus  giving  points  on  the  cube  base  curve  A  E. 
260.  The  radius  for  the  pitch  circle  of  the  cam  will  be, 


r  = 


IX  360         4.83  X  360 


2  X  TT  X  b        6.28  X  60 


=  4.62, 


where  I  =  length  of  chart  used  for  rise  of  follower  and, 
b  =  angle  during  which  the  follower  is  moving. 

With  the  above  value  of  r  the  circle  through  D  is  drawn  in  Fig. 
115.  The  arc  DEF  will  be  equal  in  length  to  the  line  DEF  in 
Fig.  114  when  drawn  to  the  same  scale,  and  it  should  be  similarly 
divided  and  the  radial  lines  at  H,  I,  .  .  .  made  equal  to  the  similarly 
lettered  ordinates  in  the  chart.  The  curve  A  E  C  thus  obtained  will 
be  the  pitch  surface  of  the  cam. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


135 


261.  EXERCISE  PROBLEM  27a.     CUBE  CURVE,  CASE  II.     Require' 
a  cube  curve  cam  with  a  circular  easing-off  arc  in  which  the  followe 


turn  of  the  cam. 

it    ( (    it     ( ( 


(a)  Rises  3  units  in  90° 

(b)  Falls  3    "     "90° 

(c)  Remains  stationary  180°     "    " 

(d)  Accelerates  during  .70  of  its  stroke. 

(e)  The  maximum  pressure  angle  to  be  30°. 


n 


75 


FIG.  115. — (Duplicate)   CUBE  BASE  CURVE  CAM,  CASE  2 

262.  TANGENTIAL  CAM,  CASE  II.  The  tangential  cam,  as  stated  in 
paragraph  218,  is  made  up  of  straight-line  sides  with  a  circular  arc 
for  rounding  off  the  end  of  the  lobe.  When  the  length  of  the  straight 
surface  of  the  cam  is  not  specified,  or  when  the  portion  of  the  stroke 
during  which  the  follower  accelerates  is  not  given  in  the  data,  the 
tangential  type  of  cam  works  out  to  good  advantage.  But  when 
either  of  the  above  items  is  included  in  the  data  for  the  tangential 
cam  it  may  conflict  with  the  proper  cam  angle  which  should  be 
allowed  for  the  follower  motion,  as  illustrated  in  the  following  prob- 
lem, which  contains  the  same  data  as  the  two  previous  problems. 
The  possible  difficulty  met  with  in  using  the  tangential  cam  arises 
from  high  accelerations  that  may  be  produced. 

^  263.  PROBLEM  28.     TANGENTIAL  CAM,  CASE  II.     Required  a  tan- 
gential cam  with  a  circular  easing-off  arc  in  which  the  follower: 

(a)  Rises  1  unit  during  60°  turn  of  the  cam. 

(b)  Falls  1    "         "     60°  "     "    "      " 

(c)  Remains  stationary  for  240°    "     "    "      " 

(d)  Accelerates  during  %  of  its  stroke. 

(e)  The  maximum  pressure  angle  to  be  30°. 


136 


CAMS 


264.  The  cam  may  be  constructed  directly  by  substituting  values 

given  in  the  data  in  the  general  formu- 
las given  in  paragraph  222,  and  then 
laying  out  the  results  as  in  Fig.  119. 
In  the  present  problem  0  A,  Fig. 
119  equals  s  as  found  in  paragraph 
222,  A  E  =  t,  0  D  =  p,  OC  =  d, 
angle  D  0  C  =  6,  angle  D  0  K  =  c, 
and  L  E  =  e.  The  radius  r  of  the 
roller  and  the  minimum  radius  w  of 
the  working  surface  are  not  shown 
in  the  illustration  but  may  be  readily 
added  if  called  for.  The  radius  of 
the  roller,  however,  cannot  be  greater 
than  EL.  The  numerical  results 

,.          ,   ,  .     ,.,     ,. 

lOUnd   by  Substituting  the  Values  given 

in  the  data  in  the  series  of  formulas 
referred  to  above  are  as  follows: 


FIG.  119.— (Enlarged)   PROBLEM  28. 
TANGENTIAL    BASE    CURVE    CAM, 

CASE  2 


=  4.84    t  =  2.79    p  =  5.58    d  =  5.84    c  =  39 


1.47. 


265.  If  it  is  desired  to  construct  the  cam  chart  for  the  purpose  of 
determining  the  velocity  and  acceleration  diagrams  later,  it  may 
readily  be  done: 

(1)  By  making  the  length  of  chart  A  R,  Fig.  118,  equal  to  the 
length  of  the  arc  D  F  on  the  cam  drawing, 


FIG.  118. — (Enlarged)  TANGENTIAL  BASE  CURVE,  CASE  2 

(2)  by  laying  off  the  pitch  line  D  F  on  the  chart  and  subdividing 
the  same  as  the  arc  D  F  on  the  cam  is  subdivided, 

(3)  by  transferring  the  radial  lines  at  //,  7,  .  .  .  from  the  cam 
to  the  chart  and  drawing  them  as  vertical  lines,  thus  obtaining  points 
for  the  base  curve  A  E  K  C. 


ADVANCED  GROUP  OF  BASE  CURVES  FOR  CAMS 


137 


266.  It  will  be  noted  that  an  attempt  to  construct  a  tangential 
cam  in  cases  such  as  the  one  here  represented  may  result  in  extremely 
large  retardation  or  acceleration  values,  as  shown  in  Fig.  121,  the 
practical  result  of  which  will  be  a  "  hard-turning  "  spot  at  a  point  on 


z> 

-3 

_^/ 

-2 

;r 

E 

1 
Y 

a 

i 

—  *l 

T 

F 

f—6 

FIG.   121. — (Duplicate)   ACCELERATION   DIAGRAM  FOR  TANGENTIAL  CAM,   CASE  2 

the  cam  corresponding  to  E,  Fig.  119,  and  continuing,  in  lessening 
degree,  to  K. 

267.  EXERCISE  PROBLEM  28a.  TANGENTIAL  CAM,  CASE  II.  Re- 
quired a  tangential  cam  with  a  circular  easing-off  arc  in  which  the 
follower: 

(a)  Rises  2  units  during  75°         turn  of  the  cam. 

(b)  Falls  2    "         "       75°  "     "    li     " 

(c)  Remains  stationary  for  210°    "     "    "     " 

(d)  Accelerates  during  .70  of  its  stroke. 

(e)  The  maximum  pressure  angle  to  be  30°. 


SECTION   VII.— CAM   CHARACTERISTICS. 

268.  METHOD  OF  DETERMINING  VELOCITIES  AND  ACCELERATIONS. 
The  velocity  and  acceleration  values  in  the  diagrams  shown  in  Figs.  72 
to  121  may  be  found  by  graphical  methods  which  are  simple  and  quite 
accurate  enough  for  most  practical  purposes  if  precision  in  drawing  is 
followed.     The  graphical  method  applies  to  all  forms  of  cams  and 
starts  with  the  cam  chart.     Its  application,  however,  is  illustrated 
only  in  connection  with  the  circular  cam  chart  in  Fig.  98,  it  being 
unnecessary  to  add  similar  lines  to  all  the  other  chart  drawings,  as 
the  constructions  would  be  the  same  in  every  case. 

269.  THE  USE  OF  TIME-DISTANCE  AND  TIME-VELOCITY  DIAGRAMS. 
The  chart  curve  A  E  C,  Fig.  98,  for  our  present  purpose,  may  be 


FIG.  98. — (Duplicate)  CIRCULAR  BASE  CURVE,  CASE  1 

termed  time-distance  curve  in  which  the  abscissa  A  R  represents 
time,  and  the  ordinates  parallel  to  A  B  represent  distances  traveled 
by  the  follower  at  corresponding  times.  If,  then,  the  time-distance 
curve  were  a  straight  inclined  line,  the  velocity  of  the  follower  would 
be  constant.  We  may  consider,  for  the  instant,  that  the  time-distance 
curve  is  straight  at  E  and  draw  a  straight  line,  E  P,  tangent  at  that 
point.  If  this  were  the  time-distance  line  and  if  it  were  continued 
for  a  time  period  represented  by  E  D,  the  follower  would  have  moved 
the  distance  P  D  in  the  time  represented  by  E  D.  If  E  D  is  consid- 

138 


CAM   CHARACTERISTICS  139 


ered  as  a  unit  of  time,  then  P  D  becomes  a  measure  of  velocity  and  its 
length  is  laid  off  in  Fig.  100,  at  XE  which  is  at  the  center  of  the  time- 
velocity  diagram.  The  length  A  C  of  the  velocity  diagram  may  be 
any  convenient  value  for  the  purpose  of  comparison.  The  distance 
D  E,  Fig.  98,  or  one-half  the  length  of  the  cam  chart,  was  selected  as 
a  time  unit  because  it  is  a  convenient  length  and  because  the  length 
of  one-half  of  each  cam  chart  represents  the  same  amount  of  time  in 
each  of  the  chart  drawings.  This  is  because  the  data  are  the  same 
in  all  the  cams  represented  in  Figs.  71  to  119.  To  find  other  points 
on  the  time-velocity  diagram,  divide  the  time-distance  curve  by  a 
number  of  equally  spaced  ordinates  as  shown  at  J,  /,  H,  Fig.  98. 
The  tangent  to  the  curve  at  K,  on  the  ordinate  J  V,  is  K  M,  and  the 
time  unit  K  L  is  equal  to  D  E.  Then,  from  the  same  reason- 
ing as  given  above  for  the  point  E,  L  M  becomes  a  measure 
of  the  velocity  of  the  follower  at  K,  and  it  is  laid  off  at  If  L  in  Fig. 
100.  Similar  constructions  are  repeated  at  the  other  points  and  the 
time- velocity  diagram  completed. 

270.  THE  TIME  ACCELERATION  DIAGRAMS  ARE  FOUND  GRAPHICALLY 
from  the  time-velocity  diagrams  by  similar  constructions.  In  Fig. 
100  a  tangent  E  S  is  drawn  to  the  time-velocity  curve  at  E  and  if  the 


— i 


FIG.  100. — (Enlarged)  VELOCITY  DIAGRAM  FOB  CIRCULAR  BASE  CURVE  CAM 

velocity  of  the  follower  is  continued  along  this  line  for  a  time  repre- 
sented by  E  Q  it  will  lose  a  velocity  of  Q  S  in  the  time  E  Q.  Such  loss 
in  velocity  is  retardation  and  consequently  the  distance  S  Q  is  laid 
off  at  E  D  at  the  center  of  the  time-acceleration  diagram  in  Fig.  101. 
The  line  E  S  in  Fig.  100  was  drawn  to  the  left,  and  consequently 
downward  to  make  the  drawing  more  compact.  In  this  way  retarda- 
tion instead  of  acceleration  was  found  logically.  Had  the  tangent 
line  E  S  been  drawn  to  the  right,  and  consequently  upward,  the 
value  Q  S  would  have  been  found  just  the  same  and  would  have  been 
called  acceleration.  The  length  of  the  acceleration  diagram,  A  C, 
in  Fig.  101  may  be  taken  any  value;  also,  the  time  unit  E  Q  in  Fig. 
100  may  be  taken  any  value  entirely  independent  of  the  time  unit 
used  in  Fig.  98,  so  long  as  the  same  length  of  line  is  taken  in  all  the 


140 


CAMS 


velocity  diagrams  as  the  time  unit,  in  making  comparisons.  If  a 
definite  speed  is  assigned  to  the  cam  then  all  the  lines  in  the  time- 
distance,  time-velocity  and  time-acceleration  diagrams  will  have  a 
definite  value  in  feet  and  in  seconds,  and  by  closely  following  these 
values,  the  diagrams  may  be  scaled  so  as  to  interpret  them  in  the 
ordinary  units  of  feet  and  seconds,  even  if  arbitrary  time  lines  have 


FIG.  101. — (Enlarged)  ACCELERATION  DIAGRAM  FOR  CIRCULAR  BASE  CURVE  CAM 


been  used  in  constructing  the  diagrams.  For  example,  suppose  that 
the  cam  in  Fig.  99  is  turning  at  120  revolutions  per  minute.  Then  it 
will  require  Vi2  second  to  turn  through  the  60°  angle  DOC,  and  D E 
in  Fig.  98  will  represent  !/24  second.  D  P  measures  .5625  inch  or 
.0469  foot.  Therefore  the  velocity  of  the  follower  at  E  will  be 


FIG.  99. — (Duplicate)   CIRCULAR  BASE  CURVE  CAM,  CASE  1 

.0469  foot  per  J/24  second,  or  1.125  feet  per  second.  The  scale  on 
XE,  Fig.  100,  would  then  be  graded  so  that  a  mark  at  1.125 
would  fall  at  E, 

In  Fig.  100,  A  C  represents  Vi2  second,  and   Q  E,  x/48  second. 
Since  X  E  represents  1.125  feet  per  second  in  this  example,  Q  S  repre- 


CAM    CHARACTERISTICS  141 

sents  .750  foot  per  second  to  the  same  scale.  Therefore  the  accelera- 
tion is  .750  foot  per  second  per  V^s  second  or  36.00  feet  per  second 
per  second.  The  scale  on  ED,  Fig.  101,  would  then  be  graded  so 
that  a  mark  at  36.00  would  fall  at  D. 

Another  set  of  construction  lines  for  obtaining  an  ordinate  in  the 
acceleration  diagram  is  shown  at  L  T  V,  Fig.  100,  where  L  T  is  the 
same  length  as  E  Q,  and  V  T  is  equal  to  the  acceleration  and  is  laid 
off  at  V  T  in  the  acceleration  diagram,  Fig.  101. 

271.  DEGREE  OF  PRECISION  OBTAINED  BY  GRAPHICAL  METHOD. 
In  Fig.  98  the  tangent  lines  may  be  drawn  with  precision  because  the 
curve  A  E  is  an  arc  of  a  circle,  but  in  the  other  curves  the  center  of 
curvature  for  each  of  the  construction  points  is  not  known  and  the 
tangent  must,  therefore,  be  drawn  by  eye.     Even  here  considerable 
precision  may  be  obtained  if,  in  so  drawing  the  tangent,  it  is  remem- 
bered that  the  tangent  at  L,  Fig.  100,  for  example,  will  be  practically 
the  same  distance  from  U  as  it  is  from  E  when  it  passes  each  of  these 
points,  provided  U  and  E  are  on  ordinates  equally  spaced,  and  pro- 
vided also  that  the  curve  A  E  has  a  fairly  uniform  rate  of  curvature 
on  both  sides  of  L.     If  the  radius  of  curvature  to  the  right  of  L  should 
grow  noticeably  shorter  than  the  radius  of  curvature  to  the  left  of  L, 
the  tangent  at  L  would  pass  a  little  closer  to  U  than  to  E.     If,  in 
addition  to  using  such  judgment  as  here  indicated  in  the  drawing  of 
tangents  to  irregular  curves,  a  sufficient  number  of  points  are  taken 
closely  together,  and  if  the  newly  derived  curve  is  drawn  smoothly 
through  the  average  positions  of  plotted  points,  a  remarkable  degree 
of  accuracy  may  be  obtained  by  the  graphical  method  of  obtaining 
velocity  and  acceleration  diagrams. 

272.  COMPARISON  OF  RELATIVE  VELOCITIES  AND  FORCES  PRODUCED 
BY  CAMS  HAVING  DIFFERENT  BASE  CURVES.     This  comparison,  which 
may  be  made  by  studying  the  several  velocity  and  acceleration 
diagrams  in  Figs.  72  to  121,  is  also  shown  in  the  accompanying  table 
where  the  maximum  velocities  of  the  follower  are  shown  in  Column  2, 
and  the  maximum  acceleration  and  retardation  values  in  Columns  3 
and  4.     Since  force  equals  acceleration  multiplied  by  mass,   the 
direct  effort  required  to  move  the  follower  is  proportional  to  the 
acceleration,   and,   therefore,  the  relative    direct    force   needed    to 
operate  the  follower  for  various  cams  is  also  shown  in  Columns  3  and  4. 
The  retardation  values  in  Column  4  represent  the  relative  pressures 
exerted  by  the  follower  against  the  cam  surface  in  slowing  up  where  a 
positive  drive  cam  is  considered.     They  also  represent  the  relative 


142 


CAMS 


sizes  of  counterweights  where  a  gravity  return  is  used.  In  the  cam 
with  the  straight-line  base  there  would  be  violent  shock  at  the  start 
and  the  cam  would  "  stick  "  and  require  considerable  direct  power, 
but  after  that  it  would  be  necessary  only  to  overcome  friction.  The 
parabola,  it  will  be  noted  from  the  table  and  from  Fig.  93,  requires 
the  least  direct  effort,  considering  the  entire  cycle  of  the  follower. 
This  effort  is  represented  by  unity  for  purpose  of  comparison.  The 
circular  base  curve  cam,  Case  II,  Fig.  113,  requires  a  trifle  less 
effort  than  the  parabola  cam  while  on  acceleration  on  the  forward 
stroke,  but  2.86  times  the  effort  of  the  parabola  while  the  follower  is 
on  acceleration  during  the  return  stroke  where  a  double-acting  cam  is 
used.  For  a  single-acting  cam  the  values  given  in  Column  4  show 
the  relative  forces  necessary  to  sufficiently  accelerate  the  follower  on 
the  return  stroke  so  as  to  keep  it  in  contact  with  the  cam. 


TABLE  SHOWING    RELATIVE    MAXIMUM    VELOCITY,  ACCELERA- 
TION AND  POWER  FOR  EACH  TYPE  OF  CAM 


FORM  OF  CAM 

RELATIVE 
MAXIMUM 
VELOCITIES 

RELATIVE  AMOUNTS  OF  DIRECT 
FORCE  NEEDED  TO  OPERATE 
CAM  DURING 

Col.  1 

Col.  2 

Acceleration 
Col.  3 

Retardation 
Col.  4 

All-logarithmic  .                ...        .... 

1.28 

1.40 
1.00 
1.31 
1.57 
2.00 
2.09 
2.16 
2.28 
2.40 
2.16 
2.79 
3.39 

1.82 

1.99 
1.25 
1.00 
1.58 
1.44 
1.60 
1.95 
0.96 
1.80 
2.55 

1.82 

1.99 
1.25 
1.00 
1.10 
1.44 
1.60 
1.95 
2.86 
4.80 
6.39 

Logarithmic  combination. 

Straight  line  

Straight-line  combination  curve  (r  =  h) 
Crank  curve 

Parabola      .                   

Tangential  curve  Case  I 

Circular  curve,  Case  I  

Elliptical  curve.       .        

Cube  curve   Case  I 

Circular  curve,  case  IT  

Cube  curve  Case  II  .         

Tangential  curve  Case  II 

273.  CAM  FOLLOWER  RETURNED  BY  SPRINGS.  Although  the  cam 
built  from  the  parabola  chart  pitch  curve  gives  the  smoothest  motion 
and  requires  the  least  direct  power  to  operate  it  so  far  as  the  cam  and 
follower  only  are  concerned,  there  may  be  other  considerations  in  the 


CAM   CHARACTERISTICS  143 

design  that  make  or  appear  to  make  some  other  form  of  chart  pitch 
curve  more  desirable.  For  example,  when  a  follower  is  returned  by  a 
positive  drive  parabola  cam,  or  when  it  is  returned  by  gravity,  the 
parabola  cam  gives  the  best  action  because  the  pull  on  the  follower 
is  constant  all  the  time,  but  when  the  follower  is  returned  by  a  spring, 
the  spring  reacts  on  the  cam  with  a  uniformly  increasing  pressure 
during  the  outstroke  as  represented  by  the  straight  inclined  dash- 
line  S  P  in  Fig.  93,  and  with  a  reverse  uniformly  decreasing  pressure 
during  the  instroke. 

274.    IF  THE  SPRING  PRESSURE  ACTING  ON  THE  CAM  IS  ZERO  when 

the  follower  is  at  rest  in  its  lowest  position,  the  spring  compression 

line  would  be  represented  by  the  straight  line  A  N,  Fig.  93,  starting  at 

A  and  inclined  so  as  to  touch  the  retardation 

line  as  at  F.      Inasmuch  as  there  should 

always  be  some  compression  in  the  spring,     -  - 

even  when  the  follower  is  at  rest,  a  margin 

of  compression  will  be  taken  as  illustrated 

at  A  S.     The  practical  spring  compression  FIG.  93.— (Duplicate)  ACCEL- 

line  will,  therefore,  be  S  P  parallel  to  A  N.      p^t7~,TT .  r!tf RAM     F°B 

'  A  ARA.BO.LA     V>AM 

As  the  follower  moves  out,  its  acceleration 

during  the  first  part  of  the  stroke  produces  increasing  pressure  between 
the  cam  surface  and  the  spring-actuated  follower  as  represented  by 
the  increasing  length  of  the  ordinates  from  S  B  to  R  D.  At  mid- 
stroke  the  follower  begins  to  slow  up.  In  the  case  shown  in  Fig.  93, 
the  slope  of  the  spring  pressure  line  was  taken  so  as  to  have  the  same 
spring  pressure  (R  F  =  S  A)  on  the  cam  at  midstroke  as  it  has  at  the 
beginning.  The  line  S  P  could  have  been  given  a  steeper  slope  if  a 
larger  margin  of  pressure  than  R  F  had  been  desired  at  midstroke. 
This  would  have  required  a  heavier  spring.  From  midstroke  to  the 
end  there  is  again  an  increasing  margin  of  pressure,  the  maximum 
being  represented  by  the  difference  between  the  ordinates  P  H  and 
R  F.  The  full  strength  of  the  spring  which  would  have  to  be  used 
would  be  represented  by  the  ordinate  P  C. 

.    275.  RELATIVE  STRENGTH  OF  SPRING  REQUIRED  FOR  CRANK,  TAN- 
GENTIAL,   CUBE    AND    PARABOLA    BASE    CURV^    CAMS.      Although    the 

parabola  cam,  with  its  perfect  action  as  described  in  paragraph  214, 
permits  of  the  use  of  a  light  spring  when  a  single  spring  is  used  to 
return  the  follower,  the  crank  curve,  tangential  curve  and  cube 
curve  cams  may  each  be  designed  to  operate  with  somewhat  lighter 
springs.  Spring  compression  lines  for  each  of  the  three  last-men- 


144 


CAMS 


tioned  cams  are  shown  at  S  P  in  Figs.  89,  97,  and  109,  and  the  max- 
imum compression  required  of  a  single  spring  in  each  case  is  1.75, 
2.35,  and  2.30  as  compared  with  2.40  for  the  parabola  cam  as  shown 
in  Fig.  93.  The  return  spring  pressure  between  the  follower  roller 
and  the  cam  surface,  when  the  crank  base  curve  is  used,  is  more 
nearly  uniform  throughout  the  entire  stroke  than  it  is  with  any  other 


89. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CRANK  CURVE  CAM 
FIG.  97. — (Duplicate)   ACCELERATION  DIAGRAM  FOR  TANGENTIAL  CAM 
FIG.  109. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CUBE  CAM 

type  of  cam,  as  may  be  noted  from  the  maximum  and  the  average 
ordinates  between  the  acceleration-retardation  curve  and  the  spring 
pressure  line,  S  P,  in  the  several  diagrams. 

276.  CUBE  CURVE  CAM  SPECIALLY  ADAPTED  for  a  follower  returned 
by  a  spring.     The  cube  curve  cam  possesses  one  characteristic  over 
the  others  in  that  the  pressure  between  the  cam  and  the  follower  is 
absolutely  uniform  during  the  latter  part  of  the  up-stroke  and  the 
first  part  of  the  down-stroke  when  the  follower  is  returned  by  a 
spring,  as  shown  by  the  parallel  lines  F  H  and  R  P,  Fig.  109.     This 
gives  an  advantage  of  smooth  running  and  uniform  wear  when  the 
spring  is  under  its  greatest  compression. 

277.  THE  PRESSURE  BETWEEN  THE  SPRING-ACTUATED  FOLLOWER 
AND  THE  CAM  is  VARIABLE  throughout  the  stroke  in  all  cams  except 
during  part  of  the  stroke  with  the  cube  curve  cam.     And  it  may 

readily  happen  that  the  acceleration  called 
for  by  the  cam  is  so  great  that  the  spring 
will  not  be  strong  enough  to  keep  the  fol- 
lower roller  against  the  cam  surface  as  may 
be  specially  noted  at  or  near  the  beginning 
of  the  return  stroke.  This  is  illustrated  in 
FIG.  113.— (Duplicate)  ACCEL-  Fig.  113  where  the  spring  pressure  against 


DIAGRAM  FOR  GIB-  the  follower  which  would  be  necessary  to 

CTJLAR    BASE     CURVE    CAM,  .  . 

CASE  2  hold  it  to  the  cam  is  represented    by  r  Hi, 

whereas,  if  a  spring  of  the  same  strength  as 

for  the  cube  .curve  cam,  Fig.  109,  were   used   the  pressure  at  the 


CAM   CHARACTERISTICS 


145 


phase  E,  Fig.  113,  would  be  only  RE.  This  means  that  the  cam 
will  "run  away  "  from  the  follower,  because  the  spring  is  not  strong 
enough  during  the  part  of  the  stroke  represented  by  T  F  R  to 
press  the  follower  against  the  rapidly  receding  cam  surface. 

278.  IN  ORDER  TO  KEEP  THE  FOLLOWER  ROLLER  AGAINST  THE 

CAM  SURFACE  where  cams  with  large  retardation  values  are  used,  as 
in  Figs.  77,  85,  113,  117,  and  121,  a  comparatively  heavy  spring  is 
required  which  will  be  unnecessarily  strong  during  a  very  large  part 
of  the  stroke,  or  else  two  springs  will  be  required,  the  second  one  to 
come  into  action  when  needed.  Both  cases  are  illustrated  in  Fig.  113. 
A  single  heavy  spring  that  will  exert  a  pressure  represented  by  W  V 


FIG.  77. 


FIG.  77. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  LOGARITHMIC-COMBINATION  CAM 
FIG.  85. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  STRAIGHT-LINE-COMBINATION  CAM 

FIG.   117. — (Duplicate)   ACCELERATION  DIAGRAM  FOR  CUBE   CAM,   CASE  2 
FIG.  121. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  TANGENTIAL  CAM,  CASE  2 

will  keep  the  follower  roller  against  the  cam  surface  at  all  times,  the 
minimum  pressure  between  the  two  occurring  at  F  G.  Or,  a  single 
and  much  lighter  spring  exerting  a  pressure  represented  by  S  P,  Fig. 
113,  may  be  used,  and  then  a  second  and  shorter  spring  with  an  initial 
compression  represented  by  M  E  may  be  so  placed  as  to  come  into 
action  at  E  so  that  the  combined  pressure  of  the  two  springs  on  the 
follower  is  M  E  plus  R  E  equal  F  E.  This  means  that  the  combined 
pressure  of  the  two  springs  will  be  just  sufficient  to  keep  the  follower 
roller  against  the  cam  at  phase  E,  and  that  the  total  pressure  of  the 
two  springs  at  the  end  of  the  stroke  will  be  represented  by  C  V,  thus 
giving  an  excess  pressure  represented  by  H  V  at  the  end  of  the  stroke. 


146 


CAMS 


The  base  curves  that  are  best  suited  for  spring-return  followers  will 
be  seen  by  comparing  the  acceleration  diagrams  in  Figs.  73  to  121 
to  be  the  crank  curve,  parabola,  tangential  curve,  Case  I,  and  the  cube 
curve,  Case  I.  The  logarithmic  combination  and  straight-line  com- 
bination curves  come  next  in  order. 

279.  ACCURACY  IN  CAM  CONSTRUCTION.  It  need  scarcely  be 
pointed  out  that  the  pitch  surfaces  of  cams  should  be  constructed 
with  considerable  accuracy  and  the  working  surfaces  carefully  fin- 


FIG.  111. — (Duplicate)  CIRCULAR  BASE  CURVE  CAM,  CASE  2 

ished,  if  definite  results  are  required,  for,  it  may  be  seen  by  com- 
paring the  pitch  surfaces  of  several  of  the  cams  illustrated  in  Figs. 
71  to  119  that  a  relatively  small  difference  in  form  may  make  a  large 
difference  in  the  velocity,  acceleration,  and  force  or  pressure,  under 
which  the  follower  operates.  For  example,  the  circular  curve  cam, 


FIG.   103. — (Duplicate)  ELLIPTICAL  BASE  CURVE  CAM 
FIG.  107. — (Duplicate)  CUBE  BASE  CURVE  CAM,  CASE  1 

Case  II,  Fig.  Ill,  and  the  cube  curve  cam,  Case  II,  Fig.  115,  are 
apparently  quite  similar  in  form,  though  varying  in  sizes,  yet  the 
maximum  accelerations  which  they  impose  on  the  follower  on  the 
return  stroke  are  quite  different,  being  2.9  and  4.8  respectively,  as 
shown  in  Figs.  113  and  117.  Also  the  cube  curve  cam,  Case  I,  Fig. 
107  and  the  elliptical  cam,  Fig.  103,  are  much  alike,  yet  their  velocity 


CAM    CHARACTERISTICS 


147 


lines  and  their  acceleration  lines,  Figs.  109  and  105,  are  different  in 
every  way  and  if  a  spring  were  used  to  return  the  follower,  the  one 
for  the  elliptical  cam  would  have  to  be  enough  heavier  to  carry  1.7 
more  compression  at  the  end  of  the  stroke  than  the  one  for  the  cube 
cam,  assuming  an  initial  pressure  of  A  S,  in  each  one.  The  value 
1.7  is  found  by  comparing  the  lengths  C  P  in  Figs.  109  and  105. 


FIG.  m: 


FIG.  109. 


FIG.  105. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  ELLIPTICAL  BASE  CURVE  CAM 
FIG.  109. — (Duplicate)  ACCELERATION  DIAGRAM  FOR  CUBE  CAM 

280.  REGULATION  OF  NOISE.  If  a  cam  follower,  as  for  example  a 
cam-operated  disk  valve,  comes  to  rest  on  a  seat  at  one  end  of  its 
stroke,  it  is  evident  that  it  would  be  desirable  for  the  follower  to  have 
the  least  possible  velocity  for  at  least  a  short  distance  before  it  reaches 
the  seat,  in  order  to  provide  against  unnecessary  striking  velocity. 
Noise  will  be  in  some  proportion  to  the  velocity  of  the  follower  at 
the  instant  of  seating.  With  this  in  mind,  an  examination  of  the 
velocity  diagrams  in  Figs.  72  to  120  will  show  that  the  cube  base 
curve,  Case  I,  Fig.  106,  gives  by  far  the  best  results,  for,  the  vertical 
ordinates  of  the  velocity  curve  in  Fig.  108  are  very  much  smaller 
as  the  follower  approaches  A  than  they  are  in  any  other  diagram, 
excepting  Case  II  of  the  cube  curve,  Fig.  116,  but  in  this  instance  the 
advantage  is  more  than  offset  by  the  high  retardation  values  at  the 
end  of  the  stroke  as  shown  in  Fig.  117.  The  circular  curve,  Case  II, 
comes  next  in  the  matter  of  giving  small  velocity  to  the  follower, 
Fig.  112,  but  it  does  not  possess  the  advantage  of  the  cube  curve 
when  a  spring  is  used  to  return  the  follower.  The  crank  curve  cam  is 
least  adapted  of  all  the  cams  where  quiet  seating  of  a  follower  is 
desired,  as  may  be  observed  by  noting  that  the  velocity  curve, 
Fig.  88,  for  this  cam  is  convex  upwards,  whereas  the  others  are 
straight  or  convex  downwards  and  thus  have  smaller  initial  vertical 
ordinates  and,  therefore,  lower  velocity.  The  full  practical  ad- 
vantage of  cams  which  give  low-seating  velocities  and  consequently  a 
more  quiet  follower  action,  is  offset  to  a  considerable  extent  where 
the  follower  operates  a  valve  which  must  admit  a  comparatively 
large  volume  of  gas  or  fluid  quickly. 


148  CAMS 

281.  HIGH  SPEED  CAMS.     Cams  intended  for  use  on  high-speed 
machines  should  give  the  smoothest  possible  motion  to  the  follower, 
that  is,  should  be  free  from  sudden  variations  of  velocity  during  the 
stroke  and  from  shock  due  to  sudden  starting  and  stopping.     A 
study  of  the  velocity  diagrams,  Figs.  72  to  120,  shows  that  the  all- 
logarithmic  and  the  straight-line  base  curves,  Figs.  72  and  80,  give 
extreme  velocity  right  at  the  start  in  all  cases;  and  that  the  logarith- 
mic-combination and  straight-line  combination  cams  will  also  give 
relatively  high  velocities  at  the  start,  Figs.  76  and  84.     Therefore 
none  of  these  cams  would,  in  general,  be  suitable  for  high-speed  work. 
Among  the  other  cams  some  have  an  advantage  at  one  end  of  the 
follower  stroke  where  the  rate  of  change  in  velocity  is  low,  but  they 
lose  it  at  the  other  end  where  it  is  high  as,  for  example,  the  cube  cam 
Case  II,  as  shown  in  Fig.  117;   or  they  lose  their  advantage  at  the 
center  or  some  intermediate  point  as  in  the  elliptical  cam,  Fig.  105. 

282.  The  cams  specified  in  the  preceding  paragraph  give  rela- 
tively large  sudden  change  of  velocity  to  the  follower  either  at  one 
end  of  the  stroke  or  the  other,  or  at  intermediate  positions;   and  of 
the  remaining  cams,  the  parabola  cam  is  the  only  one  that  gives 
absolutely  uniform  rate  of  change  of  velocity  to  the  follower.     The 
crank  curve,  the  circular  curve,  Case  I,  and  the  tangential  curve, 
Case  I,  give  relatively  good  results,  all  being  at  a  slight  disadvantage 
compared  with  the  parabola  due  to  variations  in  acceleration  of  the 
follower.     This  disadvantage,  however,  is  small,   and  these  three 
cams,   together  with  the  parabola  cam,   should  give  best  results 
where  there  is  high  speed,  provided  they  are  accurately  designed  and 
made. 

283.  BALANCING  OF  CAMS.     In  addition  to  the  forms  of  the  curves 
here  discussed  for  the  pitch  surfaces  of  cams  that  are  to  run  at  high 
speed,  it  is  necessary  to  design  the  cam  and  so  place  the  weight  that 
the  cam  will  be  as  nearly  balanced  as  possible.     This  matter  of  bal- 
ancing is  one  of  the  greatest  drawbacks  to  the  use  of  the  cam  in  high- 
speed work,  for  the  very  nature  of  a  cam  implies  irregularity  in  form 
and  hence  difficulty  in  balancing.     The  face  cam  cut  on  a  full  cir- 
cular disk  as  illustrated  in  Fig.  2  comes  nearest  to  a  natural  balance 
of  any  of  the  forms  of  radial  cams.     The  trouble  due  to  lack  of 
natural  balance  in  ordinary  radial  cams  may  easily  be  so  decided  as  to 
render  them  quite  impracticable  in  many  cases  where  high  speed 
and  large  stroke  are  required,  unless  elaborate  balancing  problems 
are  solved  in  connection  with  the  cam  design.     Small  radial  cams 
with  small  strokes  have  been  made  to  run  at  exceedingly  high  speeds. 


CAM   CHARACTERISTICS 


149 


The  cylindrical  cam,  because  of  its  natural  balanced  form  with  respect 
to  the  axis  of  rotation,  is  well  adapted  to  high  speeds. 

284.  PRESSURE  ANGLE  FACTORS  FOR  20°,  30°,  40°,  50°,  AND  60° 
FOR  VARIOUS  FORMS  OF  CAMS.     Most  of  the  base  curves  for  cams  are 
of  such  nature  that  it  is  only  necessary  to  multiply  the  follower 
motion  by  a  given  factor  and  then  multiply  the  product  by  360  and 
divide  by  the  number  of  degrees  the  cam  rotates  during  the  follower 
motion,  to  obtain  the  circumference  of  the  pitch  circle  and  the  proper 
size  of  the  cam  for  a  given  pressure  angle.     The  logarithmic  and  tan- 
gential base  curves  are  of  such  a  nature  that  no  one 'factor  can  be 
used  for  all  data  that  include  a  common  pressure  angle.     When  these 
base  curves  are  used  the  length  of  chart,  if  desired,  must  be  com- 
puted by  separate  formulas  for  each  problem.     The  logarithmic  and 
tangential  base  curves  are  most  easily  applied  by  constructing  the 
cam  pitch  surface  directly  from  calculated  values  in  each  problem 
without  the  use  of  any  chart  whatever. 

285.  The  factors  for  pressure  angles  for  all  base  curves,  excepting 
the  logarithmic   and   tangential,   are   given  in   the   accompanying 
Table  of  Factors  for  20°,  30°,  40°,  50°  and  60°.     These  factors  are 
also  laid  off  graphically  in  Fig.  132,  thus  enabling  one  to  use  inter- 
mediate values  if  desired.     For  partial  comparison  of  the  curves 
which  have  no  general  factor  with  those  which  have,  the  special  fac- 
tor in  each  case  for  the  single  comparative  problem  which  has  been 
used  throughout  in  designing  the  cams  in  Figs.  70  to  121  is  given  in 
the  following  paragraphs,  and  these  factors  are  plotted  to  give  the 
dash  lines  in  the  accompanying  chart  for  factors. 

286.  VARIED  FORMS  OF  FUNDAMENTAL  BASE  CURVES.     Several  of 
the  base  curves  are,  or  may  be,  used  in  practical  work  with  variations 


.30- 


Pitch  Line 


-2.27 


Fio,  82. — (Duplicate)  STRAIGHT-LINE  COMBINATION  BASE  CURVE 

in  details  of  construction,  as,  for  example,  in  the  straight-line  com- 
bination curve,  Fig.  82,  the  easing-off  arc  A  E  has  a  radius  A  B  equal 
to  the  total  rise  of  the  follower,  whereas  it  would  be  equally  correct 


150 


CAMS 


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CAM    CHARACTERISTICS 


151 


152 


CAMS 


in  principle  to  make  this  radius  J^  A  B.  In  this  latter  case  the  cam 
would  be  smaller  for  a  given  pressure  angle,  but  the  shock  on  starting 
and  stopping  would  be  greater.  This  case  is  not  illustrated  in  Figs. 
70  to  121  but  is  included  under  item  4  in  the  Table  of  Pressure  Angle 
Factors,  and  also  in  the  Chart  of  Pressure  Curves,  Fig.  132.  Like- 
wise the  factors  for  the  elliptical  base  curve  having  a  ratio  of  2  to  4 
instead  of  7  to  4,  are  given  in  item  5  in  the  Table  and  also  in  the 
Chart,  Fig.  132.  The  factors  for  a  cube  base  curve  made  up  of  sym- 
metrical cube  curves  are  also  given  in  item  13  in  the  Table  where  it 
may  be  noted  that  this  base  curve  gives  an  extremely  large  cam 
where  small  pressure  angles  are  desired. 

287.  METHODS  OF  DETERMINING  THE  CAM  FACTORS.     The  methods 
of  computing  the  cam  factors  for  various  base  curves  are  briefly 
described  in  the  following  paragraphs.     The  letter  h  in  the  following 
formulas  represents  the  motion  of  the  follower,  and  the  letter  a  the 
maximum  pressure  angle. 

288.  ALL-LOGARITHMIC  AND  LOGARITHMIC-COMBINATION  CURVES. 
These  base  curves  do  not  have  a  constant  factor  for  each  pressure 


Pitch  Line 


1.73- 


FIG.  78. — (Duplicate)   STRAIGHT  BASE  LINE 

angle.  The  radius  for  the  pitch  circle  in  each  problem  is  found  by 
computation  and  graphics  as  described  in  paragraph  182  et  seq. 

The  factors  for  the  data  used  in  the  charts  shown  in  Figs.  70 
and  74  are: 

For  all-logarithmic  cam:  20°,  2.28;  30°,  1.28;  40°,  .76;  50°,  .42; 
60°,  .21. 

For  logarithmic-combination  cam:  20°,  2.76;  30°,  1.69;  40°, 
1.04;  50°,  .62;  60°,  .34. 

?89.  STRAIGHT-LINE  BASE.     FIG.  78. 


1.73  =  1.73. 


CAM   CHARACTERISTICS  153 


290.  STRAIGHT-LINE  COMBINATION  BASE  CURVE,  Fig.  82. 

AR  =  2AN  +  2NX  =  2htan(-}  +hcota  =  2XlX  .268 


+  1  X  1.73  =  2.27. 

291.  CRANK  CURVE,  FIG.  86.     This  curve  may  be  regarded  as  the 
projection  of  a  helix  and,  therefore,  D  Q  equals  the  length  of  the 


FIG.  86.  —  (Duplicate)  CRANK  BASE  CUBVE 

irh.    The  line  E  Q  is  tangent 


quadrant  R  G  which  in  turn  is  equal 
to  the  base  curve  at  E. 


=  2DQXcota  =  1.57  h  cot  a  =  1.57  X  1  X  1.73  = 


2.72. 


292.  PARABOLA,  FIG.  90.  In  a  parabola,  the  subtangent  D  Q 
is  equal  to  twice  the  projected  length  of  the  curve  A  E,  and,  therefore, 
DQ  =  h 

A  R  =  2  DE  =  2  D  Q  cot  a  =  2  /*  cot  a  =  2  X  1  X  1.73  =  3.46. 


FIG.  90. — (.Duplicate)  PARABOLA  BASE  CURVE 

293.  TANGENTIAL  CURVE,  CASE  I,  FIG.  94.     This  curve  has  no 
common  factor  for  a  given  pressure  angle  and  the  radius  of  its  pitch 


FIG.  94. — (Duplicate)  TANGENTIAL  BASE  CUBVE,  CASE  2 

surface  must  be  computed  directly  by  formulas  given  in  paragraph 
222  without  the  intervention  of  a  cam  chart.  For  purposes  of  com- 
parison with  other  curves  the  following  factors  are  given ;  they  apply 


154 


CAMS 


only  for  the  data  that  have  been  used  in  the  cams  illustrated  in  Figs. 
71  to  119. 

20°,  5.28;    30°,  3.62;    40°,  2.82;     50°  2.36;    60°,  2.09. 

These  values  are  shown  in  the  dash  line  curve,  No.  9,  in  Fig.  132. 
294.  CIRCULAR  CURVE,  CASE  I,  FIG.  98.     The  chord  E  C  is  per- 
pendicular to  the  line  S  T  which  bisects  the  angle  C  S  E.    This  angle 


S 


FIG.  98.  —  (Duplicate)  CIRCULAR  BASE  CURVE,  CASE  1 

is  equal  to  the  pressure  angle.     The  line  E  F  is  perpendicular  to  C  S. 
Therefore  angle  C  E  F  equals  one-half  of  the  pressure  angle.     Then 

E  F  =  F  C  cot  H  a  and 
AR  =  2E  F  =  h  cot 


=  IX  3.73  =  3.73. 

295.  ELLIPTICAL  CURVE,  FIG.  102.     The  length  of  the  cam  chart 
for  the  elliptical  curve  for  a  pressure  angle  of  say  30°  may  be  most 


FIG.  102. — (Duplicate)  ELLIPTICAL  BASE  CURVE 

readily  found  by  constructing  several  arbitrary  elliptical  charts,  say 
four,  each  with  a  pressure  angle  factor,  or  length,  of  2,  3,  4,  and  5  re- 
spectively and  each  having  a  common  height  equal  to  the  rise  of  the 
follower.  Having  constructed  the  elliptical  curve  in  each  of  the  charts, 
draw  tangents  in  each  case  as  at  E,  Fig.  102,  and  measure  the  angle 
E  N  X  which  will  be  the  pressure  angle  corresponding  to  the  factor 
or  length  assumed.  Then,  on  any  coordinate  paper  plot  a  curve 
with  the  pressure-angle  factors  as  ordinates  and  the  corresponding 
measured  angles  as  abscissas.  This  curve  will  cross  the  ordinate 


CAM  CHAKACTERISTICS 


155 


which  passes  through  the  assigned  pressure  angle,  in  this  case  30°, 
and  the  length  of  ordinate  will  give  the  desired  cam  factor. 

296.  CUBE  CURVE,  CASE  I,  FIG.  106.     The  pressure  angle  factors 
for  this  case  in  which  two  unsymmetrical  cube  curve  arcs  are  used 


FIG.  106. — (Duplicate)   CUBE  BASE  CURVE,  CASE  1 

are  specially  computed  by  the  formulas  given  in  paragraph  242. 
The  value  of  I  in  formula  (1)  when  h  =  1,  will  give  the  factor  for 
whatever  pressure  angle  is  assigned  to  a.  For  a  pressure  angle  of  30° 

I  =  2.427  h  cot  a  =  2.427  X  1  X  1.73  =  4.20. 

297.  CIRCULAR  BASE  CURVE,  CASE  II,  FIG.  110.     The  complete 
factors  for  this  curve  are  the  same  as  for  the  circular  base  curve, 


FIG.  110. — (Duplicate)   CIRCULAR  BASE  CURVE,  CASE  2 


Case  I,  and  are  found  in  the  same  general  way.  In  Case  I  the  two  arcs 
making  up  the  base  curve  are  equal;  in  the  present  case,  they  are 
unequal,  and  the  formula  deduced  in  paragraph  294  must  be  used  for 
each  arc.  In  this  case,  the  first  circular  arc  is  required  to  lift  the 
follower  during  %  of  its  stroke,  and,  therefore,  the  distance  A  X 
in  Fig.  110  will  be, 


A  X  =  .75  h  cot 


=  .75  X  1  X  3.73  =  2.80. 


The  second  circular  arc  is  used  for  the  balance  of  the  stroke  and, 

therefore,  the  distance,  XR  =  .25  h  cot  %  a  =  .25  X  1  X  3.73  =  .93. 

298.  CUBE  CURVE,  CASE  II,  FIG.  114.    In  this  case  the  cube  curve 

is  used  for  %  of  the  stroke  and  a  circular  arc  for  the  remainder  of  the 


156 


CAMS 


stroke.     The  formula  x  =  7-* —  is  used  to  compute  the  part  A  X  of 
the  cam  chart  length.     The  value  of  h  is  the  follower's  total  motion 


,£  B 


X50fe* 


-4.83- 


>rn^ 


I 


FIG.  114.  —  (Duplicate)  CUBE  BASE  CURVE,  CASE  2 

and  that  of  /  is  the  fractional  part  of  the  follower's  motion  during 
which  acceleration  takes  place.     Then 


A  Y  — 
A  X  - 


X  .75  X  1 

>577 


_  o 
3. 


The  length  X  R  is  found  in  the  same  manner  as  in  the  preceding 
paragraph  and  is  the  same  value,  namely  .93. 

299.  TANGENTIAL  CURVE,  CASE  II,  FIG.  118.     This  curve,  like 
Case  I  of  the  tangential  cam  has  no  cam  chart,  unless  it  is  specially 


FIG.  118. — (Duplicate)  TANGENTIAL,  BASE  CURVE,  CASE  2 

desired  to  lay  it  out  after  the  cam  is  drawn  by  making  special  com- 
putations based  on  the  pitch  circle  as  described  in  paragraph  225. 
For  purposes  of  comparison  the  data  used  in  this  cam,  as  drawn  in 
Fig.  119,  are  the  same  as  for  all  other  cams  in  Figs.  71  to  119,  and  for 
the  data  so  used  the  pressure  angle  factors  are : 

20°,  13.02;    30°,  5.86;    40°,  3.36;    50°,  2.20;    60°,  1.57. 
These  values  are  shown  in  the  dash  line  curve,  16,  in  Fig.  132. 


SECTION   VIII.— MISCELLANEOUS   CAM   ACTIONS  AND 
CONSTRUCTIONS 

300.  VARIABLE  ANGULAR  VELOCITY  IN  THE  DRIVING  CAM  SHAFT. 
The  subject  of  variable  angularity  velocity  in  the  drive  shaft  of  a  cam 
applies  to  all  types  of  cams,  but  it  is  rarely  met  with  except  in  oscil- 
lating cams.     The  reason  for  this  is  that  in  machinery,  in  general, 
the  shafts  that  make  a  full  turn  do  so  with  practically  uniangular 
velocity    except    in    slow-advance    and    quick-return    motions   and 
in  some  special  cases,  and,  therefore,  the  shaft  that  operates  a  cam, 
in  general,  is  considered  to  have  uniform  angular  velocity.     But 
with  the  oscillating  cam  the  motion  must  come  through  a  crank  and 
connecting  rod,  or  eccentric  and  beam,  or  some  other  device,  from  a 
shaft  which,  in  general,  turns  with  uniform  angular  velocity,  and 
which  gives  to  the  oscillating  cam  a  variable  angular  velocity  as 
illustrated  in  Fig.  133  where  the  unequal  arcs  B\  Gi,  Gi  KI,  K\  LI 
represent  the  distances  traversed  by  the  cam  pin  B\  while  the  main- 
shaft  crank  pin  turns  through  the  equal  arcs  B  G,  G  K  and  K  L.     The 
method  of  building  a  cam  which  has  variable  angular  velocity  will  be 
illustrated  in  the  following  problem. 

301.  PROBLEM  29.     OSCILLATING  CAM  HAVING  VARIABLE  ANGULAR 
VELOCITY,  TOE  AND  WIPER  TYPE.     Required  an  oscillating  wiper  cam, 
operated  by  a  crank  and  connecting  rod  from  a  main  shaft  to  raise 
and  lower  a  straight-toe  follower  through  a  distance  of  one  unit  while 
the  crank  shaft  turns  through  120°.    Assume  the  following  dimensions : 
Main  crank  radius,  C  B,  4  units,  Fig.  133;    connecting  rod  length, 
B  BI,  20  units;  cam-arm  radius,  B\  0,  5  units;  shortest  cam  surface 
radius,  0  A,  2  units.     Find  the  distance  the  follower  will  move 
during  each  of  three  equal  periods  of  time  on  the  up-stroke. 

302.  The  first  step  in  the  solution  of  the  problem  is  to  lay  out  the 
main  crank  center  as  at  C  in  Fig.  133;  then  the  crank-pin  circle  with 
a  radius  C  B  of  4  units,  and  next  the  connecting  rod  length  of  20 
units  on  the  centerline  as  at  E  J.     Lay  off  the  assigned  120°  of  crank- 
shaft motion  symmetrically  about  the  main  centerline  as  at  BCD 
and  with  B  and  D  as  centers  and  the  length  of  the  connecting  rod 

157 


158 


CAMS 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      159 

as  a  radius  draw  two  arcs  intersecting  on  the  horizontal  centerline, 
thus  locating  B\.  With  C  as  a  center  and  the  connecting-rod  plus 
the  crank  as  a  radius,  draw  the  arc  passing  through  J;  with  C  as  a 
center  and  the  connecting  rod  minus  the  crank  as  a  radius,  draw  the 
arc  passing  through  J\. 

303.  To  find  the  center  0  of  the  cam  shaft,  Fig.  133,  take  BI  as 
a  center  and  the  assigned  cam-arm  radius  of  5  units,  and  draw  an 
arc,  on  which  the  point  0  will  be  found  later.     On  this  arc  find  a 
point,  by  trial  and  error  with  the  compass,  which  is  the  center  of  an 
arc  which  passes  through  B\  and  which  intersects  the  two  arcs  through 
J  and  Ji  at  the  same  elevation,  as,  for  example,  at  LI  and  F\.    The 
center  point  so  found  is  the  point  0.     The  arc  L\  BI  FI  will  then  be 
the  arc  of  swing  for  the  center  of  the  cam-arm  pin,  and  the  angularity 
of  action  between  the  connecting  rod  and  the  cam  arm  at  the  two 
extreme  ends  of  the  cam-arm  swing  will  be  approximately  the  same. 
Draw  a  vertical  line  through  0  and  mark  the  assigned  distance  0  A 
which  is  the  shortest  radius  of  the  cam  surface.     The  horizontal  line 
through  A  will  be  the  lowest  position  of  the  flat-surface  follower  toe. 
The  distance  A  V  is  equal  to  the  assigned  motion  for  the  follower. 

304.  Having  completed  the  general  layout  of  the  assigned  data, 
the  cam  surface  A  ¥2  is  found  as  follows :   Draw  the  arc  Bz  Lz  with 
a  radius  equal  to  0  BI,  and  make  the  length  BI  £2  equal  to  B\  L\. 
Revolve  V  about  0  until  it  meets  the  radial  line  drawn  from  L2  to  0, 
thus  determining  the  point  V\.     At  this  latter  point  draw  a  line 
V\  ¥2  perpendicular  to  0  V\.     With  the  aid  of  any  smooth-edged 
curved  ruler  draw  a  curved  line  tangent  to  A  W  at  A  and  also  tan- 
gent to  Vi  ¥2  at  the  point  where  it  happens  to  come.     Such  a  curved 
line  is  shown  at  A  ¥2  in  Fig.  133.     Any  other  curved  line  tangent 
to  the  straight  lines  A  W  and  V\  ¥2  would  have  done  the  work 
in  the  same  time  but  would  have  given  slightly  different  intermediate 
velocities  to  the  follower  as  will  be  explained  in  a  later  paragraph. 

The  actual  working  length  A  W  of  the  follower  toe  is  readily 
obtained  by  revolving  the  point  of  tangency  ¥2  about  0  until  it 
meets  the  horizontal  line  through  V  at  ¥3.  Projecting  73  down  to 
A  W  and  adding  a  short  distance  W  W\  to  prevent  a  sharp-edge 
action,  the  practical  length  A  Wi  is  obtained.  If  the  toe  shaft  is 
offset  a  distance  A  Y  the  total  length  of  follower  toe  will  be  Y  Wi. 

305.  To  find  the  distances  moved  by  the  follower  toe  during  each 
of  three  equal  periods  while  on  the  upstroke,  divide  B  L,  Fig.  133, 
into  three  equal  parts  as  at  G  and  K.    With  these  points  as  centers 


160  CAMS 

and  with  the  connecting  rod  length  as  a  radius  construct  short  arcs 
intersecting  BI  LI  as  at  Gi  and  K\.  Lay  off  the  arcs  B\  G\  and  B\  KI 
at  B2  Gi  and  B^  K%  and  draw  the  radial  lines  0  G^  and  0  K^.  Per- 
pendicular to  these  radial  lines  draw  other  straight  lines,  tangent 
to  the  curved  cam  surface  A  ¥2,  thus  obtaining  the  lines  HI  Hz  and 
/i  /2-  Revolving  HI  and  I\  back  to  the  vertical  line,  the  points  H 
and  /  will  be  obtained  and  the  distances  moved  by  the  follower  during 
the  three  equal  time  periods  on  the  upstroke  will  be  A  H,  H I  and 
/  V  respectively. 

306.  The  path  of  contact  between  the  cam  wiper  and  the  toe  is 
shown  by  the  curved  dash  line  A  V%,  Fig.  133.     Points  on  this  curve, 
such  as  at  IB,  are  obtained  by  revolving  the  point  of  tangency  /2 
around  until  it  meets  the  horizontal  line  through  7. 

307.  OTHER  CONSIDERATIONS  RELATING  TO  VARIABLE  ANGULAR 
VELOCITY  DRIVE,  brought  out  in  this  problem  (Problem  No.  29)  are 
that  the  follower  toe  takes  a  longer  time  for  the  down-stroke  as  shown 
by  the  length  of  arc  L  D  as  compared  with  L  B,  Fig.  133.     This  could 
be  rectified  and  both  times  made  the  same,  if  desired,  by  placing  the 
center  0  of  the  cam  so  that  the  points  BI  and  LI  would  be  on  the 
horizontal  line  through  C.     This  would  only  be  possible  with  certain 
limited  combinations  of  lengths  of  crank  arms  and  rods,  and  in  any 
event  the  intermediate  velocities  of  the  follower  would  be  different 
on  the  up-  and  down-strokes.     If  it  were  desired  to  know  the  distances 
moved  by  the  follower  during  three  equal  periods  on  the  down- 
stroke  the  equally  spaced  points  M  and  N,  Fig.  133,  would  be  obtained 
and  used  in  exactly  the  same  way  as  explained  for  G  and  K  in  para- 
graph 305. 

The  point  F  is  the  outward  dead  center  position  of  the  driving 
crank  pin  and  is  found  by  continuing  the  straight  line  through  FI 
and  C  to  F.  When  the  driving  crank  pin  is  at  F,  the  cam  surface 
is  in  the  position  shown  by  the  dash  line  A%  W&  and  A\  is  at  A. 
While  B  is  moving  from  D  to  F,  A\  is  moving  to  A  and  the  follower 
toe  is  at  rest,  being  supported  by  the  cylindrical  surface  A  A\  rub- 
bing against  it,  or  it  may  be  supported  by  a  resting  block  indicated 
at  ST. 

It  is  sometimes  thought  that  this  toe-and-wiper  cam  is  prac- 
tically free  from  rubbing  action  especially  where  the  length  of  the  toe 
surface  equals  approximately  that  of  the  wiper,  but  it  will  be  seen 
from  the  velocity  diagram  shown  just  above  the  cam  and  described 
in  paragraphs  317  and  318,  that  there  may  be  considerable  rubbing. 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS      161 


162  CAMS 

There  must  be  some  sliding  in  all  flat-toe  followers  where  the  acting 
surface  is  perpendicular  to  the  right-line  motion  of  the  follower,  as 
it  is  in  Fig.  133. 

308.  EXERCISE  PROBLEM  29a.     OSCILLATING   CAM  HAVING  VARI- 
ABLE ANGULAR. VELOCITY.     Required  an  oscillating  wiper  cam,  oper- 
ated by  a  crank  and  connecting-rod  from  a  main  shaft,  to  raise  and 
lower  a  straight-toe  follower  through  a  distance  of  three  units  while 
the  crank  shaft  turns  through  150°.     Find,  also,  the  distances  that 
will  be  traversed  by  the  follower  toe  during  equal  intervals  of  time  on 
the   up-stroke.     Assume    the   following    dimensions:     Main    crank 
radius,  5  units;   connecting-rod  length,  30  units;   cam-arm  radius,  7 
units;  shortest  cam  surface  radius,  4  units. 

309.  TOE-AND-WIPER    CAM    WHERE    TOE    IS    CURVED.       In    the    toe- 

and-wiper  cam  explained  in  the  paragraphs  immediately  preceding, 
a  flat  surface  toe  Y  W,  Fig.  133  was  used.  A  curved  toe  such  as  is 
shown  at  A  W,  Fig.  134  may  be  used  as  illustrated  in  the  following 
problem. 

310.  PROBLEM   30.     REQUIRED   A   WIPER   CAM   TO    OPERATE   A 
CURVED-TOE  FOLLOWER  which  shall  move: 

(a)  Up  4  units  on  the  elliptical  base  curve  where  the  ratio  of 
axes  is  2  to  4,  while  the  cam  turns  45°  in  a  counter-clockwise  direc- 
tion with  uniform  angular  velocity. 

(b)  Down  4  units  on  the  same  base  curve,  while  the  cam  turns  45° 
in  a  clockwise  direction  with  uniform  angular  velocity. 

311.  While  the  follower  toe  may  have  the  form  of  any  smooth 
curve  which  is  convex  to  the  cam  wiper,  an  arc  of  a  circle  will  be 
assumed  because  of  the  ease  in  drawing.     The  general  principles  of 
construction  are  the  same  for  this  problem  as  in  Problem  12.     The 
shortest  radius  OA  of  the  wiper  cam,  Fig.  134  is  assumed.     The 
form  of  the  curved  toe  is  the  circular  arc  A  W,  with  its  center  at  A\. 
It  is  convenient  in  such  a  problem  as  this  to  work  with  the  center 
points  of  the  follower  arc,  and,  therefore,  the  4  units  of  travel  are  laid 
off  first  at  AI  Vi  instead  of  A  V.     The  semi-ellipse  in  which  I\  Uii 
IiVi  :    :   2  :  4  is  drawn  and  the  perimeter  divided  into  equal  parts 
at  J',  Ui,  Hf.     Only  four  construction  points  are  used  in  this  problem 
in  order  to  secure  as  much  simplicity  as  possible  in  the  illustration. 
In  practice,  more  construction  points  should  be  used.     The  four 
construction  centers  at  HI,  Ii,  Ji,  V\,  are  revolved  to  their  corre- 
sponding positions  relatively  to  the  cam  at  #2,  /2,  «/2,  and  ¥2  and 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      163 


J— 

|  . 

i  ... 

*r 

- 

I 

y 

Ys 
FIG.  137.  DIAGRAM 
OF  SLIDING  VELOCIOT 

.K 

A,       V, 


FIG.   1^*4. — PROBLEM  30.     OSCILLATING  CAM  WITH  CURVED-TOE   FOLLOWER 


164  CAMS 

the  toe-arcs  drawn  as  shown  at  #3,  1 3,  Js,  and  Vs.  The  wiper  cam 
curve  A  C  is  then  drawn  tangent  to  these  arcs  and  the  tangent  points 
revolved  back  to  their  actual  positions  at  H±,  /4,  J±,  and  V±,  thus 
obtaining  the  locus  of  contact  between  the  wiper  and  toe.  This 
locus  is  shown  by  the  dashline  curve  A  H±  ¥4.  The  necessary  length 
V  YI  of  the  follower  arc  is  also  obtained  by  projecting  the  extreme 
point  Y  on  the  locus  to  FI  and  adding  an  arbitrary  distance  such  as 
FI  Wi  to  avoid  wear  at  the  tip  end. 

312.  If  an  irregular  curve  had  been  used  for  the  form  of  the  toe 
instead  of  a  circular  arc  it  would  have  been  necessary  to  construct 
a  template  of  the  desired  form  of  the  toe  and  to  move  it  out  radially 
the  desired  distances  on  each  of  the  radial  construction  lines  0  #2, 
0/2  .  .  .,  keeping  the  template  always  in  the  same  relative  position 
with  each  of  the  radial  lines.     At  each  of  the  four  adjustments  of  the 
template,  arcs  would  have  been  drawn  against  the  template  edge 
and  the  work  then  continued  as  described  in  the  preceding  para- 
graph. 

313.  The  pressure  angles  in  the  toe-and-wiper  cams  are  quite 
different  for  flat  and  curved  toes.     In  Fig.  133  the  line  of  pressure 
is  always  parallel  to  the  axis  Y  FI,  of  the  follower  rod,  as  illustrated 
by  the  vertical  line  at  W  V% ;  and  the  maximum  leverage  with  which 
it  acts  on  the  bearings  is  Y  W.     With  the  curved-toe  wiper  the  line 
of  pressure  is  an  inclined  line  and  the  pressure  angle  at  the  top  of 
the  stroke  is  V  V\  V±,  Fig.  134,  and  when  the  follower  is  half  way  up 
the  pressure  angle  is  1 I\  1 4- 

314.  EXERCISE  PROBLEM  30a.     REQUIRED  A  WIPER  CAM  TO  OPER- 
ATE A  CURVED-TOE  follower  which  shall  move: 

(a)  Up  3  units  with  uniform  velocity  while  the  cam  turns  60°  in  a 
counter-clockwise  direction  with  uniform  angular  velocity. 

(b)  Down  3  units  with  uniform  velocity  while  the  cam  turns  60° 
in  a  clockwise  direction  with  uniform  angular  velocity. 

315.  RATE  OF  SLIDING  OF  CAMS  ON  FOLLOWER  SURFACE.     The 
rubbing  velocity  of  cams  which  are  in  sliding  contact  with  the  fol- 
lower, may  be  readily  determined  by  constructing  simple  velocity 
diagrams  at  each  of  the  construction  points,  as  explained  in  the  fol- 
lowing paragraphs. 

316.  PROBLEM  31.     RATE  OF  SLIDING  BETWEEN  CAM  AND  FLAT 
FOLLOWER  SURFACES.     Find  the  curve  of  rubbing  velocity  between 
surfaces  in  a  toe-and-wiper  cam  mechanism  where  the  follower  toe  is 


MISCELLANEOUS    CAM    ACTIONS   AND    CONSTRUCTIONS      165 

a  flat  surface.     Assume  that  the  wiper  oscillates  with  uniform  angular 
velocity. 

317.  In  Fig.  135  let  the  angle  /4  0 1 5  represent  the  uniform  angular 
velocity  of  the  wiper  cam.  Then  the  point  1 2  on  the  cam  will  have 
the  linear  velocity  /4 1 5.  Laying  this  value  off  at  1%  IQ,  where  /2 

Fio.  186— DIAGRAM  OF  VELOCITIES 


FIG.   135. — PROBLEM  31. — SLIDING  IN  TOE-AND-WIPER  CAMS 


comes  into  action,  and  taking  the  component  1%  /?,  the  actual  rubbing 
velocity  is  obtained.  This  may  be  transferred  to  1%  Ii  in  Fig.  136 
and,  finding  other  ordinates,  the  complete  sliding-velocity  curve 
A\  V-j  is  obtained.  The  ordinate  A  A\  is  quickly  obtained,  for  it  is 
obviously  equal  to  the  linear  velocity  line  A  A\  in  Fig.  135.  In 
Fig.  135  the  detail  construction  for  obtaining  the  velocity  of  the  fol- 
lower is  shown  only  at  one  point,  7 3,  the  construction  at  the  other 


166  CAMS 

points  being  the  same.  Also  all  lines  pertaining  to  the  construction 
of  the  cam  are  omitted,  as  they  are  fully  given  in  Fig.  52. 

318.  THE  ACTUAL  RATE  OF  SLIDING  IN  FEET  PER  SECOND  may  be 
readily  found  at  any  position  by  means  of^the  velocity  diagram 
in  Fig.  136.     For  example,  if  the  cam  shaft  0,  Fig.  135,  is  consid- 
ered to  oscillate  back  and  forth  through  45°,  100  times  per  minute 
with  uniform  angular  velocity,  and  if  the  radius  0  I±  is  14  inches,  the 
line  /4  Id  will  be  drawn  to  represent  a  velocity  of 

2  X  14  X  3.14  X  2  X  100 

8  X  12  X  60  =  3'°4  feet  Per  SeC°nd' 

This  value,  laid  off  as  the  resultant  velocity  at  1 3,  gives  a  component 
or  sliding  velocity  1%  li  which  is  laid  off  at  1%  Ii  in  Fig.  136.  Other 
ordinates,  found  in  the  same  way,  will  give  the  curve  A\  TV,  showing 
the  sliding  velocity  between  cam  and  toe  in  feet  per  second.  The 
minimum  rate  of  sliding  will  be  A  A\  shown  in  both  Figs.  135  and  136, 
and  will  be  1.6  measured  on  the  same  scale  that  was  used  to  lay  out 

hi  5. 

319.  THE  VELOCITY  OF  THE  FOLLOWER,  in  feet  per  second,  may 
also  be  readily  found  by  simply  taking  the  vertical  component  IB  Is, 
Fig.  135,  and  laying  it  off  at  1%  Is  in  Fig.  136.     Taking  the  vertical 
components  at  other  points  the  line  A  Vs,  showing  the  linear  velocity 
of  the  follower  will  be  obtained.     The  line  A  Vs  is  a  straight  line  in 
this  problem  because  this  cam  illustration  was  taken  so  that  the 
follower   would   have   uniformly   increasing   velocity.     In   general, 
where  the  cam  curve  A  ¥2  in  Fig.  135  is  assumed,  the  line  A  Vs  in 
Fig.  136  will  not  be  straight. 

320.  EXERCISE  PROBLEM  3 la.     SLIDING  VELOCITY  BETWEEN  CAM 
AND  FOLLOWER.     Assume  a  flat-toe  follower  with  a  rise  of  3  inches 
and  a  cam  wiper  of  minimum  radius  of  4  inches  which  oscillates  with 
uniform  angular  velocity  through  150  cycles  per  minute.     Construct 
the  toe-and-wiper  surfaces  and  find  the  curve  of  sliding  velocity 
between  them  in  feet  per  second.     Find  also  the  curve  of  linear  veloc- 
ity of  the  follower  toe  and  state  the  maximum  velocity  in  feet  per 
second. 

321.  PROBLEM  32.    SLIDING  VELOCITY  WITH  CURVED  TOE  FOL- 
LOWER.    Find  the  curve  of  rubbing  velocity  between  surfaces  in  a 
curved  toe-and-wiper  cam  mechanism,   assuming  that  the  wiper 
oscillates  with  uniform  angular  velocity. 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      167^ 

In  curved-toe  followers  the  general  principle  of  obtaining  the 
rubbing  velocity  is  the  same,  although  the  detail  of  drawing  the 
velocity  diagram  differs  slightly.  In  Fig.  134  the  linear  velocity  of 
the  paint  H%  on  the  cam  is  H$  HQ  and  this  value  is  laid  off  at  H±  HI. 
The  direction  of  sliding  at  this  phase  must  be  that  of  the  common 
tangent  line  to  the  two  surfaces,  and  its  length,  which  represents  the 
velocity  of  sliding,  is  found  by  drawing  the  line  H?  HS  parallel  to  the 
direction  of  motion  of  the  point  H±  on  the  follower.  The  length  of 
H±  HS  is  thus  found  and  is  laid  out  as  shown  in  Fig.  137,  directly  over 
H±  of  Fig.  134.  Other  lines  representing  the  rubbing  velocity  are 
similarly  found  and  laid  out  in  Fig.  137,  thus  obtaining  the  rubbing 
velocity  curve  AiH&  Vs. 

322.  In  the  case  of  the  curved-toe  follower  it  will  be  noted  that 
that  portion  of  the  toe  from  ¥4  to  YI,  Fig.  134,  will  be  traversed 
twice  as  often  as  the  portion  from  V  to  V±,  and  in  addition  the  rubbing 
velocity  will  be  much  greater.     In  the  flat-toe  follower,  Fig.  135,  the 
point  of  contact  travels  regularly  forth  and  back  the  full  distance  on 
each  stroke,  but  the  wear  as  in  the  curved-toe  follower  will  be  irreg- 
ular, due  to  the  variable  rubbing  velocity,  which  in  the  case  illustrated 
in  Fig.  136  is  a  maximum  at  the  tip  end. 

323.  EXERCISE  PROBLEM  32a.     SLIDING  VELOCITY  WITH  CURVED- 
TOE  FOLLOWER.     Find  the  curve  of  rubbing  velocity  between  cam 
surfaces  in  Problem  30a,  assuming  that  the  wiper  cam  oscillates 
through  a  cycle  90  times  per  minute.     Show  scale  for  curve. 

324.  PROBLEM  33.     SLIDING  VELOCITY  WHERE  CAM  HAS  VARIABLE 
ANGULAR  VELOCITY.     Find  the  curve  of  rubbing  velocity  between 
surfaces  in  a  flat  toe-and-wiper  cam  construction,  assuming  that  the 
wiper  cam  oscillates  with  a  variable  angular  velocity. 

325.  When  an  oscillating  cam  has  variable  angular  velocity,  as 
in  Fig.  133,  the  extent  of  the  sliding  action  between  cam  and  follower 
may  be  found  as  in  the  present  example.     In  Fig.  133,  the  length 
of  crank  represented  by  C  E  is  4  inches  and  the  crank  is  assumed  to  be 
turning  120  revolutions  per  minute.     The  velocity  of  the  crank  pin 

•n  *u      u2  X  3-14  X  4  X  120          in  , 
will  then  be  -         10       An         -  =  4.19  feet  per  second. 

\.£i  X  OU 

326.  The  velocity  just  obtained  is  represented  by  the  line  K  U, 
Fig.  133,  laid  off  to  any  convenient  scale.     Its  component  K  Ui  along 
the  rod  is  found  by  dropping  from   U  a  perpendicular  to  the  con- 
necting-rod position  K  K\.     The  component  K  Ui  is  then  trans- 
ferred to  the  other  end  of  the  rod  at  K\  Uz.     This  component  gives  a 


168  CAMS 

resultant  linear  velocity  of  K\  U%  to  the  cam  crank  pin  at  the  phase  K\. 
At  the  radial  distance  0  K^,  which  is  equal  to  the  radii  01  2,  and  0  Is 
the  linear  velocity  will  be  K%  t/4  and  this  transferred  to  1%  will  give 
/3  C/5  as  the  resultant  linear  velocity  of  /2  when  it  becomes  the 
driving  point.  The  line  1%  UQ  is  the  component  in  the  direction  in 
which  sliding  must  take  place  and  this  is  laid  off  at  7s  U&  in  Fig.  138. 
If  K  U  represents  4.19  feet  per  second,  7s  UQ,  will  represent  1.30  feet 
per  second  to  the  same  scale  and  the  maximum  velocity  of  sliding, 
which  is  represented  at  A%  A&,  will  be  1.87  feet  per  second. 

327.  EXERCISE  PROBLEM  33a.     SLIDING  VELOCITY  WHERE  CAM 
HAS  VARIABLE  ANGULAR  VELOCITY.     Assume  crank  C  E,  in  Fig.  133, 
to  be  5  units  long  and  turning  at  rate  of  100  revolutions  per  minute; 
also,  take  the  angle  B  C  D  =  150°  symmetrical  about  C  E,  the  con- 
necting rod  B  BI  —  30  units,  the  cam  arm  0  B\  =  7  units,  the  mini- 
mum cam  radius  0  A  =  4  units  and  the  cam  lift  3  units.     Construct 
the  cam  and  follower  and  draw  the  curve  of  sliding  velocity  to  scale. 

328.  ELIMINATION  OF  SLIDING  FRICTION  WHERE  FLAT  OR  CURVED 
SURFACE  FOLLOWERS  ARE  USED.     The  ordinary  toe-and-wiper  cam 
mechanism  operates  with  more  or  less  sliding  action  as  shown  in  the 
preceding  paragraphs.     Cams   resembling   the   toe-and-wiper  type 
may  be  constructed  so  as  to  eliminate  all  sliding  friction  by  using 
special  curves  and  lines  for  the  wiper  and  toe  surfaces  as  will  be 
explained  in  succeeding  paragraphs.     Fig.  139  shows  a  straight  sur- 
face toe  moving  up  and  down  in  a  straight  line  while  in  Fig.  146  a 
similarly  moving  toe  has  a  curved  working  surface.     In  both  there  is 
pure  rolling  action.     Likewise,  in  Figs.  142  and  145  the  working 
surface  of  the  follower  arm  is  straight  in  one  case  and  curved  in  the 
other,  yet  in  both  cases  there  is  pure  rolling  action.     In  all  cases  of 
pure  rolling  action  on  flat  or  curved  surfaces  it  is  impossible  to  assign 
various  intermediate  velocities  to  the  follower  as  part  of  the  data  of 
the  problem. 

329.  THE  PRINCIPLE  OF  PURE  ROLLING  ACTION  BETWEEN  CAM 
SURFACES.     It  is  a  fundamental  principle  of  pure  rolling  action 
between  two  rotating  surfaces  that  the  point  of  contact  between  them 
must  always  be  on  the  line  of  centers.     This  is  illustrated  in  Fig.  141 
where  the  point  of  contact,  C,  is  on  the  line  of  centers  A  B,  and  where 
the  contact  point  between  the  curves  C  D  and  C  E  will  always  be  on 
the  line  of  centers.     This  principle  also  applies  in  Fig.  139,  where  the 
follower  toe  B  D  is  moving  up  and  down  in  a  straight  line  and  where 
it  must  be  considered  that  the  toe  is  turning  about  a  point  on  the  line 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      169 

A  B  F  at  an  infinite  distance.  Then  A  F  becomes  the  line  of  centers 
and  the  point  of  contact  between  B  C  and  B  D  will  always  be  on  the 
line  B  F. 

330.  WELL-KNOWN   CURVES   THAT   LEND   THEMSELVES   READILY 
TO  PURE  ROLLING  ACTION  in  cam  work  are  the  logarithmic  spiral  and 
the  ellipse.     Examples  of  these  will  be  given  in  following  paragraphs, 
where  the  solutions  are  entirely  graphical  and  comparatively  simple. 
The  parabola  and  the  hyperbola  may  also  be  readily  used  for  rolling 
cam  surfaces.     Any  line  or  curve  that  may  readily  be  expressed  by  a 
mathematical  equation  may  also  be  taken  as  one  surface  and  the 
equation  for  the  other  curve  that  will  work  with  it  in  pure  rolling 
action  may  be  derived.     An  example  of  this  is  given  in  paragraph  346. 
The  use  of  the  logarithmic  curve  for  pure  rolling  action  in  the  toe- 
and-wiper  type  of  construction  where  the  follower  toe  has  a  straight- 
edge working  surface  and  moves  in  a  straight  line  is  given  in  the 
paragraphs  immediately  following. 

331.  PROBLEM  34.     PURE  ROLLING  WITH  FLAT  SURFACE  FOLLOWER. 
Required  an  oscillating  logarithmic  cam  arm  that  will  give  a  straight- 
line  reciprocating  motion  to  a  flat-surface  follower  arm,  with  pure 
rolling  action: 

(a)  The  follower  to  move  up  4%  units,  while  the  cam  turns  30°. 

(b)  The  pressure  angle  to  be  20°. 

332.  This  problem  is  illustrated  in  Fig.  139  where  the  flat-surface 
toe  B  D  is  moved  from  the  solid-line  position  to  the  dash-line  position 
while  the  cam  ABC  swings  through  the  angle  C  A  F.     The  method 
of  constructing  the  problem  is  as  follows : 

Draw  the  horizontal  line  A  F,  Fig.  139,  and  from  any  point  B 
draw  a  line  BD  making  an  angle  with  BF  equal  to  the  assigned 
pressure  angle.  Continue  B  D  until  the  vertical  distance  between  it 
and  B  F  is  equal  to  the  assigned  lift  of  the  follower,  4J4  units  in  this 
problem  as  measured  at  D  F.  Mark  the  point  F.  Assume  the  dis- 
tance A  B  sufficient  to  allow  for  the  cam  shaft  and  cam  hub.  A  B 
is  taken  as  4  units  in  this  problem,  and  A  F  is  found  upon  measuring, 
to  be  16  units.  Substitute  these  values  in  the  following  general 
equation : 

180°  X  tan a        R  _  180°  X  .364  _ 

,rX  .434     log  7  ^  -314^:434- 

in  which  r  =  4,  R  =  16,  a  =  20°,  and  in  which  9  gives  the  angle 
whose  limiting  radial  line  A  C  is  equal  in  length  to  A  F: 


170 


CAMS 


333.  The  angle  of  28.8°  is  then  laid  off  at  F  A  C  as  shown  in  Fig. 
139  by  means  of  a  protractor.  If  a  protractor  is  not  at  hand  this 
angle  may  be  readily  constructed  with  the  aid  of  a  trigonometrical 
table  from  which  the  tangent  of  28.8°  is  found  to  be  .55.  Lay  off 
A  E  equal  to  one  unit  on  any  independent  scale  and  draw  a  perpen- 
dicular line  EHatE.  On  this  line  lay  off  .55  of  this  unit  thus  obtain- 
ing the  point  H.  The  angle  E  AH  will  then  be  28.8°.  Draw  A  H 
and  continue  it  to  A  C  making  A  C  =  A  F  =  16  units  on  the  scale 


FIG.  139. — PROBLEM  34,  OSCILLATING  CAM  WITH  PURE  ROLLING  ACTION  ON  FLAT  SUR- 
FACE FOLLOWER 

of  the  cam  drawing.  The  logarithmic  curve  through  B  and  C  will 
be  the  one  which  will  work  in  pure  rolling  action  with  the  straight 
line  B  D. 

334.  To  obtain  other  points  on  the  curve  B  C  as  at  J,  assume 
intermediate  values  for  R  in  the  above  formula,  r  remaining  the  same 
as  before.  Taking  R  at  14  units  and  again  solving  the  equation,  0 
is  found  to  be  26°  and  this  angle  is  laid  off  at  F  A  J.  A  J  is  made 
14  units  in  length.  In  like  manner  other  points  shown  by  dots 
between  J  and  B  may  be  found  by  taking  R  equal  to  12,  10,  8,  and  6 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      171 

in  successive  computations  and  laying  off  the  resulting  angles  which 
are  found  to  be  22.9°,  19.1°,  14.4°,  and  8.45°  respectively. 

335.  The  pressure  angle  between  the  two  cam  surfaces  will  be  a 
constant  and  equal  to  a.     The  smaller  the  pressure  angle,  the  longer 
will  be  the  toe  of  the  follower  for  a  given  lift.     As  a  corollary  to  the 
conditions  of  pure  rolling  action  it  follows  that  the  developed  length 
of  the  logarithmic  arc  B  C  must  be  equal  to  the  length  of  the  straight 
line  B  D.     The  stem  E  G  of  the  follower  toe  may,  in  general,  be  taken 
with  its  center  line  midway  between  B  and  F. 

336.  EXERCISE  PROBLEM  34a.     PURE  ROLLING  WITH  FLAT  SUR- 
FACE FOLLOWER.     Required  an  oscillating  cam  arm  that  will  give  a 
straight-line  reciprocating  motion  to  a  flat-surface  follower  arm, 
with  a  pure  rolling  action : 

(a)  The  follower  to  move  up  1  unit  while  the  cam  arm  turns  30°. 

(b)  The  pressure  angle  to  be  15°. 

337.  THE  USE  OF  THE  LOGARITHMIC  CURVE  FOR  PURE  ROLLING 
ACTION  between  two  rolling  cam  arms,  where  both  arms  oscillate,,  is 
shown  in  the  paragraphs  immediately  following.     Before  taking  up  a 
definite  problem  it  is  necessary  to  consider,  in  order  to  obtain  a 
satisfactory  understanding,  some  of  the  properties  peculiar  to  the 
logarithmic  curve.     These  properties  are: 

First.  That  in  a  series  of  equally  spaced  radial  lines  drawn  from 
the  pole  to  the  logarithmic  curve,  the  length  of  any  one  line  is  a  mean 
proportional  of  the  lines  on  either  side.  To  illustrate,  the  curve  G  H, 
Fig.  140,  is  a  logarithmic  curve,  the  radial  lines  A  G,  A  K,  A  L,  and 
A  H  are  spaced  by  equal  angles  and  A  K  :  A  L  :  :  AL  :  A  H, 
or,  A  L  =  V  A  K  X  A  H.  The  spacing  angle  may  be  of  any  size. 

Second.  That  the  difference  in  length  between  any  two  radial 
lines  drawn  from  the  pole  to  the  curve  will  be  the  same  no  matter 
where  those  radial  lines  are  taken,  providing  they  intercept  equal 
lengths  of  arc.  To  illustrate  the  difference  C\  A  —  E  A,  Fig.  140, 
is  equal  to  D  A  —  C  A  for  the  reason  that  the  arcs  C  D  and  E  Ci, 
were  made  equal  in  developed  length. 

Third.  That  a  tangent  and  a  radial  line  at  any  point  on  a  loga- 
rithmic curve  form  the  same  size  of  angle,  no  matter  where  the  point 
is  taken.  To  illustrate,  the  angle  between  the  tangent  Q  C  and 
the  radial  line  A  C,  Fig.  140,  equals  the  angle  between  Qi  H  and  A  H. 

338.  PROBLEM  35.     PURE  ROLLING  WITH  LOGARITHMIC  CURVED 
CAM  ARMS.     Construct  a  pair  of  pure  rolling  oscillating  cam  arms 


172 


CAMS 


with  logarithmic  curved  surfaces,  the  driver  swinging  through  21° 
and  the  follower  arm  through  one-half  of  that  angle. 

339.  The  first  step  in  the  solution  of  Problem  35  consists  in  draw- 
ing a  logarithmic  curve  of  any  desired  curvature  by  assuming  any 
convenient  angle  such  as  60°  as  shown  at  K  A  H  in  Fig.  140  and  any 
two  lengths  of  lines  as  shown  at  A  K  and  A  H.  K  and  H  will  then 


FIQ.  140. — PROBLEM  35,  BASIC  LOGARITHMIC  CURVE  FOR  OSCILLATING  CAM  ARMS  HAV- 
ING PURE  ROLLING  ACTION 

be  points  on  the  logarithmic  curve.  To  find  an  intermediate  point, 
bisect  the  angle  K  A  H  as  at  A  L  and  make  A  L  a  mean  proportional 
between  A  K  and  A  H  in  accordance  with  the  first  general  principle 
of  paragraph  337.  To  find  a  point  to  the  left  of  K,  make  the  angle 
K  A  G  equal  to  angle  L  A  K',  then  A  K  becomes  the  mean  propor- 
tional, and  A  G  :  A  K  :  :  A  K  :  A  L,  or  A  G  =  ^-. 

A.  LJ 

340.  Having  constructed  the  general  logarithmic  curve  as  above, 
lay  off  an  angle  of  21°  with  the  vertex  at  A,  Fig.  140  and  with  the 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      173 

sides  at  A  C  and  A  D,  or,  the  sides  may  be  in  any  other  position, 
according  to  the  length  desired  for  the  cam  arm.  Make  a  tracing  of 
of  the  angle  CAD  and  of  the  arc  C  D  and  reproduce  it  at  C  A  D  in 
Fig.  141.  Also  draw  the  body  outlines  of  the  cam  arm,  and  the  driver 
is  then  completed.  To  find  the  follower,  step  off  the  arc  C  D,  Fig. 
140,  into  four  or  six  steps  with  the  dividers  and  restep  the  distance  C  D 
off  on  another  part  of  the  logarithmic  curve  where  the  newly  placed 
arc,  equal  to  C  D,  will  be  subtended  by  an  angle  of  10^°  as  specified 
in  the  data.  The  new  position  for  the  length  of  the  arc  must  be  found 
by  trial,  and  in  this  problem  it  is  at  E  Ci  in  Fig.  140  where  the  arc 
E  Ci  equals  C  D  in  length  and  the  angle  E  A  C\  equals  one-half  of 
CAD.  The  angle  E  A  C\  and  the  arc  E  C\  are  now  traced  on  tracing 
cloth  and  redrawn  at  C  B  E  in  Fig.  141.  Upon  drawing  the  outlines 
for  the  arm  the  follower  is  completed. 


FIG.  141. — SWINGING  CAM  ARMS  WITH  LOGARITHMIC  SURFACES  IN  PURE  ROLLING  ACTION 

341.  THE  ANGULAR  MOTION  OF  EACH  CAM  depends  on  the  positions 
on  the  logarithmic  curve  at  which  the  equal  arcs  are  taken.     Had  it 
been  desired  to  swing  the  shaft  B  through  a  larger  angle,  the  loga- 
rithmic arc  E  Ci,  Fig.  140,  would  have  been  taken  lower  down.    When 
E  Ci  coincides  in  position  with  C  D,  the  arm  C  B,  Fig.  141,  will 
swing  through  the  same  angle  as  the  arm  C  A  and  both  arms  will  be 
of  the  same  length  and  identical  in  every  wa,y, 

342.  TANGENCY  OF  LOGARITHMIC  CAM  SURFACES.     The  fact  that 
the  two  rolling  cam  curves  C  D  and  C  E,  Fig.  141,  are  tangent  at  C 
follows  from  the  third  principle  laid  down  in  paragraph  337,  which 
^points  out  that  the  tangents  at  C  and  Ci,  Fig.  140,  make  the  same 

angles  with  C  A  and  Ci  A  respectively.  Since  C  and  C\  come 
together  on  the  same  straight  line  A  B  in  Fig.  141,  the  angle  B  C  Qi 
in  that  figure  will  equal  the  angle  A  C  Q. 

343.  REGULATION    OF    PRESSURE    ANGLE    WHERE    LOGARITHMIC 
ROLLING  CAMS  ARE  USED.     Logarithmic  curves  of  varying  sizes,  or 
expansion,  may  be  used  for  rolling  cam  surfaces,  but  in  general,  the 


174  CAMS 

best  results  will  be  obtained  by  using  curves  having  a  large  expansion. 
The  expansion  may  be  measured  specifically  by  noting  the  rate  of 
increase  in  the  length  of  the  successive  radial  lines  which  are  drawn 
at  equal  angles  with  each  other.  The  greater  the  expansion  of  the 
logarithmic  curve,  the  smaller  will  be  the  pressure  angle,  or  radial 
pressure  on  the  bearings  of  the  cam.  This  is  shown  in  Fig.  141, 
where  P  C  R  is  the  pressure  angle  and  C  S  is  the  radial  pressure  on 
the  bearings.  If  curve  C  D  had  a  greater  expansion,  its  normal 
C  R  would  fall  nearer  C  P  and  the  pressure  angle  would  be  smaller. 

344.  EXERCISE  PROBLEM  35a.     PURE  ROLLING  LOGARITHMIC  CAM 
ARMS.     Construct  a  pair  of  rolling  oscillating  cam  arms  with  curved 
surfaces,  the  driver  swinging  through  an  angle  of  30°,  and  the  fol- 
lower through  20°. 

345.  DERIVED  CURVE  FOR  ROLLING  CAM  ARMS.     Any  line  or  curve 
that  is  expressed  by  a  mathematical  equation  may  be  taken  as  the 
form  of  an  oscillating  cam  arm,  and  the  equation  for  another  curve 
that  will  work  with  it  in  pure  rolling  action  may  be  derived.     In 
the  paragraphs  immediately  following,  a  cam  arm  with  a  straight 
surface  is  assumed  and  the  curve  that  will  work  with  it  is  derived. 
The  derivation  of  the  curve  involves  the  use  of  calculus,  but  the 
results  are  comparatively  easy  to  apply  practically. 

346.  PROBLEM  36.     THE  USE  OF  A  DERIVED  CURVE  FOR  ROLLING 
CAMS.     Given  a  straight-edge  oscillating  follower  arm.     Required  a 
curved  oscillating  arm  that  will  work  with  it  with  pure  rolling  action. 

347.  In  solving  the  above  problem  the  following  notation,  illus- 
trated in  Fig.  142,  will  be  used  :  a  =  angle  between  the  line  of  centers 
of  the  oscillating  arms  and  the  line  of  the  straight  follower  surface 
for  the  phase  in  which  this  line,  when  extended,  passes  through  the 
axis  of  the  driver  shaft.     The  angle  a  is  a  constant. 

b  =  angle  turned  through  by  the  straight  follower  arm,  at  any 

phase,  measured  from  the  horizontal  position. 
c    =  construction  angle  for  the  curved  follower  arm. 
L  =  distance  between  centers. 
R  =  working  radius  of  follower  arm  at  any  phase. 
S  —  working  radius  of  driver  arm  at  phase  corresponding  to  R. 

Then,  sin  a  =  --    =  -;     ..........     (1) 


7?  = 


sinBDC      sin  b' 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS      175 

..     (3) 

,.. 
TT  X  0.4343     g  sin  %  (b  -  a) 

Assuming  L  =  24,  and  Ri  =   4,  we  have  from  equation  (1), 


L-  R; 

180°  tan  a        cos  ^  (6  +  a) 


sin  a  =  —  =  0.1668,  and  from  a  table  of  sines,  a  = 

,W 


FIG.  142. — PROBLEM  36,  SWINGING  CAM  ARMS  WITH  DERIVED  SURFACES  IN  PUBE  ROLLING 

ACTION 


Then,  from  equations  (2)  and  (3),  for 
4 


6  =  12°,  R 


b  =  15°,  R  = 


b  =  20°,  R  = 


.208 


.2588 


.342 


=  19.22  and  S  =  24  -  19.22  =  4.78 


=  15.45 


8  =  24  -  15.45  =  8.55 


=  11.70    "     S  =  24  -  11.70  =  12.30 


Similarly,    for    b  =  30°,    R  =  8.00;     for    b  =  50°,    R  =  5.23;     for 
bt=  70°,  R  =  4.26;  and  for  b  =  90°,  R  =  4  =  Ri. 
From  equation  (4),  for 

180°  X  .1718         cos  y2  (12°  + 
3.1416  X  .4343     '  sin  %  (12°  - 


22.67  log  -          =  22.67  X  1.70  =  38.6°; 


176 


CAMS 


b  =  15°,  c  =  22.67  log  '~  =  22.67  X  1.32 


30°; 


b  =  20°,  c  =  22.67  log 


22.67  X  1.033  =  22.8 


Similarly  for  6  =  30°,  c  =  16.5°;  for  6  =  50°,  c  =  9.1°;  for  6  =  70°. 
c  =  4.2°;  and  for  6  =  90°,  c  =  0°. 

348.  Plotting  the  above  values  of  R  in  Fig.  142,  B  H  =  19.22  and 
A  H  =  4.78;  B  J  =  15.45;  and  B  D  =  11.70,  etc.  A  test  of  the 
accuracy  of  the  work  thus  far  may  now  be  made  by  drawing  a  line 
from  H  tangent  to  the  circle  having  Ri  for  a  radius  and  noting  if  it 
makes  an  angle  of  12°  with  the  line  of  centers.  Likewise  a  line  from 


FIG.  142. — (Duplicate)  PROBLEM  36,  SWINGING  CAM  ARMS  WITH  DERIVED  SURFACES 
IN  PURE   ROLLING  ACTION 


D  tangent  to  this  same  circle  should  make  an  angle  of  20°  with  D  B, 
etc. 

Again,  plot  the  values  of  c,  starting  with  any  phase  in  which  it 
is  desired  to  show  the  cams.  In  this  case  the  phase  illustrated  is 
for  the  straight  cam  at  an  angle  of  20°.  Lay  off  the  angle  D  A  P  = 
22.8°.  Then,  starting  with  A  P  as  a  datum  line  lay  off  the  values 
of  c  as  found  above,  making  the  arc  P  Q  =  38.6°,  arc  P  T  =  30°,  etc. 

Finally  on  each  of  the  lines  A  Q,  A  T,  etc.,  lay  off  the  correspond- 
ing values  of  S.  These  values  have  already  been  found  to  be  4.78 
and  8.55  respectively,  etc.  Thus  the  points  F,  V,  D  .  .  .  P  on  the 
follower  cam  curve  are  obtained. 


MISCELLANEOUS    CAM   ACTIONS   AND    CONSTRUCTIONS      177 

349.  ROLLING  CAMS  USEFUL  FOR  STARTING  SHAFTS  GRADUALLY. 
The  curve  F  D  P  is  tangent  to  the  circular  arc  having  A  P  for  a 
radius.     This  suggests  an  interesting  and  perhaps  useful  mechanical 
addition  in  that  gear  teeth  might  be  cut  on  G  P  W  as  a  pitch  line,  and 
also  on  Z  C  N'  as  a  pitch  line,  thus  permitting  an  oscillating  shaft  A  to 
give  a  certain  number  of  complete  revolutions  in  opposite  directions 
to  the  shaft  B.    In  starting  each  cycle,  the  shaft  B  would  accelerate 
gradually,  and  it  would  come  to  rest  gently  at  the  end  of  its  cycle. 
The  rate  at  which  the  motion  of  the  shaft  B  would  accelerate  at  start- 
ing is  indicated  by  the  ratios  ^FH  to  ^r^.     Giving  the  actual  values 

ti  £>  J\  £> 

which  these  ratios  have  in  this  problem,  it  is  found  that  the  accelera- 
tion of  B  increases  from     '  22  to  -r-  or  from  about  ]/±  of  the  angular 

velocity  to  five  times  that  of  the  shaft  A . 

350.  REGULATION  OF  PRESSURE  ANGLE  WITH  DERIVED  ROLLING 
CURVE.     Returning  to  Problem  36  and  considering  it  only  as  a  cam 
mechanism  it  will  be  noted  that  the  angles  taken  for  b  in  the  compu- 
tations become  the  pressure  angles  and  show  a  measure  of  the  radial 
thrust  that  goes  into  the  bearings  without  producing  any  useful 
rotative  effort.     For  example,  in  Fig.  142,  the  cams  are  in  contact 
at  D  and  the  normal  pressure  is  represented  by  D  U.     The  component 
pressure  D  X  goes  to  the  bearings  and  D  Y  is  useful  in  turning  the 
shaft  B.     The  pressure  angle  U  D  Y  is  20°.     When  G  reaches  M,  it 
will  be  in  contact  with  Z  and  the  pressure  angle  will  be  50°. 

351.  EXERCISE  PROBLEM  36a.     THE  USE  OF  A  DERIVED  CURVE  FOR 
ROLLING   CAMS.      Given  a  straight-edge   oscillating  follower  arm. 
Required  a  curved  oscillating  arm  that  will  work  with  it  in  pure 
rolling  action.     Let  distance  between  centers  of  oscillating  arms 
be  30  inches  and  the  maximum  theoretical  length  of  the  curved  toe 
20  inches. 

352.  ELLIPTICAL  ARCS  FOR  PURE  ROLLING  CAMS.     Pure  rolling 
oscillating  cam  arms  having  arcs  of  ellipses  for  their  working  sur- 
faces, may  also  be  used.     In  constructing  these  cams,  use  is  made  of 
the  characteristic  of  the  ellipse  that  the  sum  of  the  two  lines  drawn 
from  any  point  on  the  perimeter  to  the  foci  will  be  constant  and 
will  be  equal  to  the  length  of  the  major  axis.     Briefly  then,  it  is  only 
necessary  to  take  two  identical  ellipses  and  center  them  on  one  pair 
of  their  foci  at  a  distance  apart  equal  to  their  major  axis.     Such  a 
pair  of  ellipses,  illustrated  at  A  C  and  B  D  in  Fig.  143,  will  then 


178  CAMS 

turn  through  360°  respectively  and  will  be  in  pure  rolling  contact  all 
the  time.  Oscillating  rolling  cam  arms  may  be  obtained  from  the 
ellipses  by  simply  taking  equal  and  symmetrically  placed  arcs  from 
each  as  shown  at  E  F  and  GH,  Fig.  143.  The  following  problem 
will  illustrate  the  method  of  construction. 

353.  PROBLEM  37.     PURE  ROLLING  ELLIPTICAL  CAM  ARCS,  ANGLES 
OF  ACTION  EQUAL.     Construct  a  pair  of  oscillating  rolling  cam  arms 
whose  working  surfaces  are  arcs  of  ellipses.     Take  the  distance 
between  centers  as  24  units,  make  the  angle  of  action  of  the  driver 
and  follower  shafts  the  same,  and  find  the  pressure  angle  at  any 
point. 

354.  In  constructing  Problem  37,  lay  down  the  assigned  dis- 
tances between  centers,  24  units,  as  at  A  and  B  in  Fig.  143.     These 
points  will  lie  at  the  fixed  focus  of  each  ellipse.     Take  any  point, 
such  as  K,  on  the  line  of  centers  between  A  and  B.     The  nearer 
K  is  taken  to  one  of  the  foci  the  smaller  will  be  the  pressure  angles 
between  the  rolling  cam  surfaces  according  to  this  construction, 
other  data  being  the  same.     With  K  as  a  center  and  K  B  as  a  radius 
draw  an  arc  B  L  of  any  desired  length  thus  obtaining  the  angle  B  K  L 
which  may  be  any  value.     The  smaller  it  is  taken  the  flatter  will  be 
the  resulting  working  surface  G  H  of  the  cam  and  the  smaller  will  be 
the  pressure  angles.     Had  K  been  taken  midway  between  A  and  B, 
and  had  the  angle  B  K  L  been  made  90°,  a  limiting  case  would  have 
resulted  in  which  the  ellipses  from  which  the  cam  arms  are  taken 
would  have  had  a  minimum  eccentricity  and  the  cam  arms  would 
have  had  the  largest  angle  of  action,  but  the  pressure  angles  would 
have  been  larger.     With  K  as  a  center  draw  the  arc  A  I  making 
angle  A  K I  equal  to  B  K  L.    Then  L  and  I  are  the  free  foci  of  the 
basic  ellipses. 

355.  TO    FIND    THE    MAJOR    AND    MINOR    AXES    OF    THE    ELLIPSES 

take  L  and  A,  Fig.  143,  as  centers  and  one-half  of  A  B  as  a  radius 
and  draw  short  arcs  intersecting  at  M  and  at  N  as  indicated  at  M. 
Also  use  B  and  I  as  centers  in  the  same  way,  thus  obtaining  0  and  P. 
M  and  N  will  then  be  the  extremities  of  the  minor  axis  of  one  ellipse, 
and  0  and  P  the  extremities  of  the  other.  From  J,  which  is  midway 
between  A  and  L,  lay  off  distances  J  Q  and  J  C  equal  also  to  one-half 
of  A  B.  Q  and  C  are  then  the  extremities  of  the  major  axis  of  one 
ellipse,  and  similarly  D  and  R  are  the  extremities  of  the  other  ellipse. 

356.  To  FIND  POINTS  OF  THE  ELLIPSE  take  a  piece  of  paper,  or 
a  thin  card,  having  a  perfectly  straight  edge  as  indicated  by  the  dash 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      179 

and  double-dot  lines  in  Fig.  143.  Mark  the  points  T  and  U  on  the 
edge  of  the  paper  a  distance  apart  equal  to  the  semi-minor  axis  0  S, 
and  also  mark  the  point  V  so  that  its  distance  from  T  is  equal  to  the 


x 


II 


E- 


C 


w 


/p 


..Fio.  143. — PROBLEM  37,  BASIC  ELLIPSES  FOR  PURE  ROLLING  CAM  ARMS,  ANGLES  OP 

ACTION  EQUAL 

semi-major  axis  D  S.  Then  move  the  paper  so  as  to  keep  the 
point  U  always  on  the  major  axis,  and  V  always  on  the  minor  axis, 
and  the  point  T  will  move  in  the  path  of  the  desired  ellipse. 

357.    To  OBTAIN  AN  EQUAL  ANGLE  OF  ACTION  FOR  BOTH  ELLIPTICAL 

CAMS,  as  called  for  in  the  statement  of  the  problem,  equal  lengths 


180  CAMS 

of  arcs  symmetrical  about  the  extremity  of  the  minor  axis  are  taken 
from  each  ellipse.  Thus  2V  E  equals  N  F,  Fig.  143,  and  0  G  equals 
0  H  equals  N  F.  The  angle  of  action  for  each  cam  is  then  equal  to 
E  A  F.  This  angle  may  be  made  larger  or  smaller  by  increasing  or 
decreasing  the  arcs  N  F  and  N  E.  These  arcs,  however,  should 
not  approach  too  closely  to  the  extremities  of  the  major  axis,  for  the 
pressure  angle  then  increases  rapidly,  as,  for  example,  when  the  con- 
tact point  moves  from  F  toward  Q. 

358.  PRESSURE  ANGLE  IN  ROLLING  ELLIPTICAL  CAM  ARMS.    The 
pressure  angle  is  the  angle  between  the  perpendicular  to  the  radial 
line  at  the  point  of  contact  and  the  normal  to  the  curve  at  that  point. 
It  varies  at  different  phases  and  is  a  minimum  when  the  extremities 
of  the  minor  axes  are  in  contact,  that  is  when  N  and  0,  Fig.  143, 
come  in  contact  at  X.     Consequently  the  angle  S  0  Z  is  the  pressure 
angle  when  0  is  in  action.     The  line  0  Z  is  perpendicular  to  the  radial 
line  B  0,  and  the  line  0  S  is  normal  to  the  curve  at  0.    At  K  the 
angle  W  K  Y  is  the  pressure  angle.     The  normal  to  the  ellipse  at 
any  point,  such  as  K  Y,  may  be  readily  found  by  making  use  of  the 
property  of  the  ellipse  that  the  normal  to  the  curve  at  any  point  is 
the  bisector  of  the  angle  formed  by  the  focal  lines  from  that  point. 
For  example  K  B  and  K  I  are  focal  lines  from  K,  and  K  Y  bisects 
the  angle  B  K  I. 

359.  EXERCISE  PROBLEM  37 a.     PURE  ROLLING  ELLIPTICAL  CAM 
ARMS,  ANGLES  OF  ACTION  EQUAL.     Construct  a  pair  of  oscillating 
rolling  cam  arms  whose  working  surfaces  are  ares  of  ellipses.     Take 
the  distance  between  centers  as  20  units,  make  the  angle  of  action 
of  each  shaft  the  same,  and  find  the  pressure  angle  at  the  extremity 
of  the  angle  of  action. 

360.  PROBLEM  38.    ELLIPTICAL  ROLLING  CAM  ARCS,  ANGLES  OP 
ACTION  UNEQUAL.     Construct  a  pair  of  oscillating  rolling  cam  arms 
whose  working  surfaces  are  composed  of  an  arc  of  an  ellipse.     Take 
the  distance  between  centers  as  24  units,  make  the  angle  of  action  of 
the  driver  2.9  times  that  of  the  follower  and  find  the  maximum 
pressure  angle. 

361.  In  the  solution  of  this  particular  problem  any  ellipse  may  be 
used  whose  major  axis  is  24  units  long.     The  shorter  the  minor  axis 
is  taken  the  less  will  be  the  pressure  angle,  and  the  smaller  also  will 
be  the  actual  practical  angles  through  which  the  cam  arms  will 
swing.     In  laying  down  the  problem  take  Q  C,  Fig.  144,  equal  24 
units,  as  the  major  axis  of  the  ellipse.    Bisect  Q  C  at  X,  and  assume 


MISCELLANEOUS   CAM    ACTIONS   AND   CONSTRUCTIONS      181 

X  M  and  X  N  as  the  semi-minor  axes.  With  M  and  N  as  centers 
and  a  radius  equal  to  Q  X  draw  short  arcs  intersecting  the  major  axis 
at  A  and  at  L.  These  points  will  be  the  foci  of  the  ellipse.  Construct 
the  ellipse  as  directed  in  paragraphs  355  and  356.  Select  an  arc  of  such 
length  and  position  on  the  ellipse  that  it  will  subtend  focal  angles,  i.e. 


FIG.  144. — ANGLES  OF  ACTION  FOB  ELLIPTICAL  CAM  ARMS 

angles  whose  vertices  are  at  the  foci,  which  are  to  each  other  as  1  is  to 
2.9.  Such  an  arc  is  shown  at  F  E  and  it  subtends  an  angle  of  24° 
from  the  vertex  at  L  and  an  angle  of  70°  from  the  vertex  at  A.  The 
value  of  2.9  given  in  the  data  is  now  provided  for  because  70  divided 
by  24  equals  2.9.  The  arc  F  E  is  used  for  the  form  of  the  working 
surface  of  the  two  cam  arms,  as  directed  in  the  following  paragraph. 
362.  To  construct  the  cam  arms  for  Problem  38,  lay  down  the 
shaft  centers  by  marking  the  points  A  and  B,  Fig.  145,  24  units  apart. 
On  a  piece  of  tracing  paper  draw  the  arc  F  E  of  Fig.  144  and  mark 
the  point  A.  Lay  the  tracing  paper  down  in  Fig.  145  with  A  of  the 


H 


Fia.  145. — PROBLEM  38,  PURE  ROLLING  ELLIPTICAL  CAM  ARMS,  ANGLES  OF  ACTION 

UNEQUAL 

tracing  at  A  of  the  figure,  and  with  E  of  the  tracing  on  the  center 
line  A  B.  Redraw  the  traced  curve  in  Fig.  145,  giving  E  F.  Again, 
on  the  tracing  paper  draw  the  curve  F  E  of  Fig.  144  and  mark  the 
point  L.  Lay  the  tracing  paper  down  in  Fig.  145  with  L  of  the 


182  CAMS 

tracing  at  B,  and  E  of  tracing  on  the  center  line  A  B.  Redraw  the 
traced  curve  in  Fig.  145,  giving  EH.  E F  and  E H  should  be  tan- 
gent at  E  if  the  work  is  correctly  done.  The  forms  of  the  arms  and 
the  hubs  are  now  drawn,  and  the  angles  of  70°  and  24°  are  indicated 
as  shown  in  Fig.  145,  thus  giving  a  ratio  of  turning  angles  of  2.9  to  1 
as  required.  The  maximum  pressure  angle  will  be  at  the  point  on 
the  working  curve  that  is  nearest  to  the  extremity  of  the  major  axis 
of  the  original  ellipse,  and  it  will  be  equal  to  50°  as  shown  at  W  F  Y, 
Fig.  144. 

363.  EXERCISE  PROBLEM  38a.     ELLIPTICAL  ROLLING  CAM  ARCS, 
ANGLES  OF  ACTION  UNEQUAL.     Construct  a  pair  of  oscillating  rolling 
cam  arms  whose  working  surfaces  are  composed  of  an  arc  of  an  ellipse. 
Take  the  distance  between  centers  as  18  units,  make  the  angle  of 
action  of  one  shaft  2.2  times  that  of  the  other,  and  find  the  maximum 
pressure  angle. 

364.  PURE  ROLLING  PARABOLIC  CAM  SURFACES  FOR  A  RECIPRO- 
CATING MOTION.     The  parabola  lends  itself  to  pure  rolling  action  in 
cam  work,  but  it  can  be  used  only  when  either  the  driver  or  the  fol- 
lower has  rectilinear  motion,  and  then  the  rectilinear  motion  must 
be  in  a  direction  perpendicular  to  the  line  of  axes  of  the  two  para- 
bolas when  they  are  in  contact  at  their  vertices. 

365.  PROBLEM  39.     ROLLING  PARABOLAS.     Required  a  parabolic 
oscillating  cam  to  give  rectilinear  motion  to  the  follower  with  pure 
rolling  action.     Assume  the  length  of  path  of  contact,  and  find,  (a) 
angle  of  action  of  driver,  (6)  range  of  motion  of  follower,  and,  (c)  the 
maximum  and  minimum  pressure  angles. 

366.  CONSTRUCT  A  PARABOLA  by  making  use  of  the  property  that 
a  point  on  the  curve  is  equidistant  from  the  focus  and  the  directrix. 
To  do  this,  assume  a  point  A,  Fig.  146,  as  the  focus  of  the  parabola 
on  the  line  X  X  as  an  axis.     Assume  the  directrix  F  F  at  right 
angles  to  the  axis  and  at  any  desired  distance  A  B  from  the  focus. 
The  larger  A  B  is  taken  the  larger  will  be  the  oscillating  cam  for  a 
given  motion,  and  the  smaller  will  be  the  pressure  angles.     The 
vertex  of  the  parabola  will  be  at  J  midway  between  A  and  B.     A 
point  on  the  curve  may  be  found  by  assuming  any  radius,  such  as 
A  D,  and  drawing  a  short  arc  as  shown  at  D  using  A  as  a  center; 
then  laying  off  this  radial  distance  on  the  axis  starting  from  the  direc- 
trix as  at  B  Di,  and  drawing  a  perpendicular  line  Di  D  until  it  meets 
the  arc  at  D.     The  point  D,  thus  obtained,  will  then  be  a  point  on 
the  parabola  and  other  points  may  be  found  in  the  same  way  and  the 


MISCELLANEOUS   CAM    ACTIONS   AND    CONSTRUCTIONS       183 

curve  S  J  V  drawn.  Draw  a  radial  line  A  K  perpendicular  to  the 
desired  direction  of  motion  of  the  follower,  which  direction,  in  this 
problem,  is  C  W.  On  this  radial  line,  assume  any  distance  R  K 
as  the  path  of  action;  then,  arcs  of  circles  through  R  and  K  having  A 
for  a  center  will  limit  the  part  G  E  of  the  parabolic  curve  which  will 
form  the  driver  cam  surface.  The  entire  oscillating  cam  G  E  A  may 
now  be  drawn. 


x  Dl 

RK  =  Path  of  Action 


FIG.  146. — PROBLEM  39,  PARABOLIC  CAM  SURFACES  FOR  PURE  ROLLING   RECIPROCATING 

MOTION 

367.  The  surface  M  N  on  the  follower  arm,  Fig.  146,  will  be 
identical  with  G  E,  already  constructed,  and  may  be  readily  found  by 
drawing  the  lines  S  J  V  and  X  X  on  a  piece  of  tracing  cloth  or  paper, 
turning  the  paper  on  the  reverse  side,  and  then  adjusting  it,  always 
keeping  X  X  of  the  tracing  parallel  to  A  K,  until  the  two  curves  come 
tangent  as  shown  at  C.  The  axis  X  X  of  the  tracing  will  then  be  in 
the  position  Z  Z.  The  follower  cam  surface  and  rod,  M  N  U  may 
now  be  drawn.  The  angle  of  action,  range  of  action,  and  pressure 
angles  may  now  be  found  as  indicated  in  Fig.  146. 


184  CAMS 

The  two  parabolic  surfaces  GE  and  M  N,  Fig.  146,  will  be  in 
pure  rolling  action  on  the  path  K  R,  the  driving  cam  turning  about  A, 
and  the  follower  cam  moving  in  a  direction  perpendicular  to  K  A . 

368.  PURE  ROLLING  HYPERBOLIC  CAM  ARMS  WHERE  CENTERS  ARE 
CLOSE   TOGETHER.     Two   equal   hyperbolas   will   give   pure   rolling 
action  to  two  oscillating  cam  arms,  the  essential  features  of  con- 
struction being  that  the  hyperbolas  should  turn  about  one  pair  of 
foci  as  fixed  centers  and  that  the  distance  between  these  centers 
should  equal  the  distance  between  the  vertices  of  the  hyperbolas. 
In  Fig.  147  A  and  B  are  the  foci  of  one  hyperbola  and  C  0  and  D  U 
its  two  branches,  while  H  and  S  are  the  foci  of  the  other  hyperbola 
and  V  W  one  of  its  branches. 

369.  PROBLEM  40.  ROLLING  HYPERBOLAS.     As  a  problem  illus- 
trating the  application  of  hyperbolas  to  rolling  cam  work,  let  it  be 
required  to  construct  two  cam  arms  and  shafts  and  determine  the 
angle  of  action  of  eacl}.  and  the  maximum  and  minimum  pressure 
angles. 

370.  CONSTRUCT  THE  HYPERBOLAS  by  making  use  of  the  property 
that  the  distances  from  any  point  on  the  curve  to  two  fixed  points, 
called  the  foci,  have  a  common  difference.     Therefore,  assuming  A 
and  B  as  foci,  Fig.  147,  and  C  as  a  vertex,  the  common  difference  to 
be  used  throughout  will  be  C  B  minus  C  A,  equals  C  D.     By  assuming 
different  distances  between  A  and  B,  and  A  and  C,  different  angles  of 
action  and  different  pressu  e  angles  will  be  obtained. 

371.  A  point  on  the  hyperbola,  such  as  E,  Fig.  147,  is  found  by 
taking  any  radius  such  as  B  E  and  striking  an  arc  with  B  as  a  center. 
Then  with  A  as  a  center  draw  another  short  arc  with  a  radius  equal 
to  B  E  minus  C  D.     Where  the  second  arc  crosses  the  first  will  be 
a  point  on  the  curve  as  at  E.     Other  points  are  found  in  the  same 
way. 

372.  The  center  of  one  shaft  will  be  located  at  the  focus  B  if  it  is 
desired,  for  example,  to  show  the  cam  surfaces  in  contact  on  the 
branch  C  0.     Assuming  E  as  the  point  of  tangency  of  the  two  cam 
surfaces,  the  center  of  the  other  shaft,  and  consequently  one  focus 
of  the  other  hyperbola,  must  be  on  the  line  E  B}  for  the  reason  that 
in  pure  rolling  action  the  point  of  contact  must  always  be  on  the  line 
of  centers. 

373.  The  second  hyperbola  must  be  placed  with  respect  to  the 
first  so  that  the  distances  between  the  fixed  foci  and  the  free  foci  are 
equal  to  each  other  and  to  the  distance  between  the  vertices  of  the 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS       185 

hyperbola.  Therefore,  take  the  distance  CD  and  lay  it  off  at 
B  H,  also  use  it  as  a  radius  with  A  as  a  center  to  draw  the  short  arc 
through  S.  With  H  as  a  center  and  A  B  as  a  radius  draw  another 
arc  intersecting  the  first  at  S,  thus  determining  the  second  focus,  S, 
of  the  second  hyperbola  and  its  axis  if  it  is  desired.  The  tangent 
branch  V  W  may  now  be  independently  constructed  as  explained 
for  C  0,  or  it  may  be  traced  from  C  0  of  which  it  is  a  duplicate. 


FIG.  147. — PROBLEM  40,  HYPERBOLIC  CAM  SURFACES  FOR  SWINGING  ARMS  ON  CLOSE 
CENTERS  WITH  PURE  ROLLING  ACTION 

374.  The  path  of  action,  assuming  K  G  to  be  the  driving  cam 
surface,  the  angles  of  action  for  both  cam  shafts,  and  the  maximum 
and  minimum  pressure  angles  may  be  obtained  from  a  study  of  the 
illustration  in  Fig.  147. 

Rolling  hyperbolic  cams  differ  from  all  others  in  that  the  path  of 
action  may  lie  entirely  on  one  side  of  the  centers  of  rotation,  instead 
of  between  the  centers  as  is  the  case  with  the  logarithmic  and  ellip- 


186  CAMS 

tical  arms.  The  practical  value  of  this  is  that  it  will  permit  driving 
action  between  two  shafts  that  are  very  close  together,  as  indicated 
by  the  shafts  H  and  B  in  Fig.  147. 

375.  DETAIL  DRAWING  OF  CYLINDRICAL  CAMS.     A  simple  practical 
method  for  constructing  the  surface  guide  line  for  the  center  of  the 
cutting  tool  in  cylindrical  cams  was  explained  in  paragraph  127.     A 
more  elaborate  method  of  construction  giving  a  more  precise  mechan- 
ical action  and  a  more  complete  drawing  of  the  cam  is  now  given. 

376.  TO  FIND  THE  TRUE  MAXIMUM  PRESSURE  ANGLE  OF  A  CYLIN- 
DRICAL CAM,  the  pitch  cylinder  and  not  the  surface  cylinder  should 
be  drawn  first.     The  pitch  cylinder  is  shown  at  Br  W  H 3  .  .  .  $3, 
Fig.  148  and  is  drawn  with  a  radius  of  5.2,  the  data,  excepting  for 
pressure  angle,  being  the  same  as  for  Problem  15.     Briefly  the  data 
are:    (a)  Follower  to  move  in  a  straight  line  4  units  to  the  right  on 
the  crank  curve  while  the  cam  turns  120°;    (b)  To  move  to  left  4 
units  on  crank  curve  while  cam  turns  120°;    (c)  To  dwell  while  cam 
turns  120°;   (d)  the  true  maximum  pressure  angle  to  be  30°. 

The  pitch  curve  Q%  PI  J,  etc.,  is  then  obtained  and  the  normal 
J  Gi  is  drawn  as  in  Problem  15.  This  normal  will  make  an  angle  of 
30°  with  J  D  if  the  work  is  correctly  done.  This  is  the  assigned 
maximum  pressure  angle  and  is  at  the  bottom  of  the  pin;  the  pressure 
angle  D  J  G  at  the  surface  of  the  cylinder  will  be  less  than  30°. 
In  this  problem  the  data  and  layout  were  such  that  the  point  J  of 
maximum  pressure  angle  could  be  readily  made  to  fall  on  the  front 
element  of  the  cylinder  and  the  angle  D  J  Gi  thus  shown  in  its  true 
size.  Where  the  data  and  layout  are  such  as  not  to  conveniently 
bring  the  pitch  point  on  the  front  element  of  the  cylinder,  the  pitch 
curve  will  have  to  be  revolved  if  it  is  desired  accurately  to  show  the 
pressure  angle  in  true  size  on  the  drawing. 

377.  DRAWING  OF  GROOVE  OUTLINES  FOR  CYLINDRICAL  CAM.     If 
it  is  desired  to  draw  the  groove  outlines  of  a  cylindrical  cam  one  of  two 
methods  may  be  used,  (1)  the  approximate  method,  or,  (2)  a  more 
exact  method.     The   approximate  method   which   is   simpler   and 
quicker  and  which  gives  good-appearing  results  where  the  slope  of 
the  groove  does  not  exceed  30°  is  applied  by  laying  off  the  points  1 
and  2,  Fig.  148,  at  equal  distances  on  each  side  of  7/2  on  the  surface 
pitch  curve,  these  points  representing  the  extremities  of  a  diameter  of 
the  follower  pin.     Similarly  the  points  3-4,  5-6,  etc.,  are  obtained. 
A  curve  drawn  through  the  points  S,  1,  3,  5,  etc.,  will  represent  one 
of  the  surface  edges  of  the  groove.     The  bottom  lines  of  the  groove 


MISCELLANEOUS    CAM    ACTIONS   AND    CONSTRUCTIONS        187 


are  found,  for  example,  by  projecting  J$  to  J±  and  then  laying  out 
the  diametral   line  9-10.     A  curve  through  9  and  other  similarly 


5° 


*      GO 


found  points  will  represent  one  of  the  bottom  edges,  and  the  curve 
through  10,  etc.,  will  represent  the  other. 

378.  A  MORE  EXACT  METHOD  OF  DRAWING  THE  OUTLINES  OF  THE 

GROOVE  consists  in  drawing  the  projection  of  a  section  of  the  pin 


188  CAMS 

which  is  tangent  to  the  cylinder.  The  section  will  appear  in  general 
as  an  ellipse  in  the  side  view  and  the  curves  representing  the  groove 
edges  will  be  drawn  tangent  to  these  ellipses  instead  of  through  the 
extremities  of  the  major  axes  as  described  in  the  preceding  paragraph. 
The  detail  work  for  this  is  shown  in  Fig.  148  where  the  straight  line 
7 '-8'  is  the  end  projection  of  a  pin  section  which  is  tangent  to  the 
cylinder.  The  points  7  and  8  are  projected  from  7'  and  8'  and  are 
the  extremities  of  the  minor  axis  of  the  ellipse;  the  horizontal  line, 
5—6  passing  through  Ji  is  equal  to  the  diameter  of  the  pin  and  is 
equal  to  the  major  axis.  /2  is  projected  from  J'.  The  ellipse 
5,  7,  6,  8,  is  now  constructed,  as  shown.  Similar  ellipses  should  be 
constructed  at  other  points  as  at  /2,  #2,  etc.  At  A  the  ellipse 
becomes  a  circle  and  at  E  it  flattens  to  a  straight  line.  The  curve 
S  £2  drawn  tangent  to  the  ellipses  instead  of  through  the  extremities 
of  the  major  axes  will  be  one  of  the  surface  edges  of  the  groove. 
Even  with  this  refined  method  of  construction  there  remains  an 
approximation,  for  it  will  be  evident  that  the  circle  at  the  top  of  the 
pin  lies  in  a  plane  which  is  tangent  to  the  cylinder  and  that  the  pro- 
jection of  this  circle  gives  an  ellipse  that  does  not  lie  on  the  surface 
of  the  cylinder.  Therefore,  the  curve  drawn  tangent  to  the  ellipse 
would  not  lie  on  the  cylinder.  The  error,  however,  in  following  the 
above  directions  is  too  small  to  show  in  a  drawing  of  practical  pro- 
portions. If  desired,  this  slight  error  in  construction  may  be  cor- 
rected by  rounding  off  the  end  of  the  pin  to  conform  with  the  curve 
of  the  cylinder  and  projecting  the  curve  of  the  rounded  end  of  the 
pin  to  the  side  view  to  give  the  elliptical-like  curve  to  which  the 
groove  curve  is  tangent.  This  is  illustrated  in  a  case  of  exaggerated 
proportions  in  Fig.  149  where  M  is  a  true  ellipse  and  is  a  projection  of 
the  end  of  the  pin  when  it  is  flat.  N  is  a  projection  of  the  perimeter 
of  the  pin  when  its  end  is  rounded  off  to  conform  with  the  curve  of 
the  cylinder. 

379.  FORMS  OF  FOLLOWER  PINS  FOR  CYLINDRICAL  CAMS.  Cylin- 
ders, cones  and  hyperboloids  may  be  used  for  the  form  of  follower 
pins  to  work  in  the  groove  of  cylindrical  cams.  A  cylindrical  pin, 
drawn  to  a  large  scale,  is  shown  at  G  J  in  Fig.  149  lying  in  a  groove 
which  is  cut  in  the  cam  cylinder  C  Z.  The  cylindrical  pin  is  advanced 
longitudinally  the  distance  E  F,  Fig.  149,  while  the  cam  turns  through 
the  angle  A\  0\  A 5,  Fig.  150.  The  top  edges  of  the  groove  are  rep- 
resented by  the  helical  curves  G  G±  and  H  H±,  Fig.  149,  and  the  bot- 
tom lines  of  the  working  side  surfaces  of  the  groove  are  represented 


MtSCELLANEOUS    CAM    ACTIONS   AND    CONSTRUCTIONS        189 

by  7  /4  and  J  J±.  The  center  of  the  follower  pin  is  shown  in  its 
central  position  at  A 2-  The  straight  line  Az'Gz  is  a  normal  to  the  top 
pitch  line  A  A 2  A 4  of  the  groove,  and  it  is  the  line  of  pressure  between 
the  side  of  the  groove  and  the  follower  pin  at  the  surface  of  the 
cylinder.  The  angle  K  A%  0  is  the  pressure  angle  at  the  top  of  the 
pin  and  it  is  made  30°  in  this  example  as  shown  in  Fig.  149.  The 
straight  line  A  2  1 2  is  a  normal  to  the  helix  B  A  2  B±  which  is  the  locus 
of  the  center  point  of  the  bottom  of  the  follower  pin.  The  line  A 2  1 2, 
then,  is  the  line  of  pressure  between  the  side  of  the  groove  and  the 
pin  at  the  bottom  of  the  pin,  and  L  A<z  0  is  the  pressure  angle  at  the 
bottom  of  the  pin.  The  pressure  angle,  therefore,  varies  from  the 
top  to  the  bottom  of  the  groove,  being  smallest  at  the  top.  From 
this  it  follows  that  the  pitch  surface  of  a  cylindrical  cam  should  be 
at  the  shortest  radius  reached  by  the  follower  pin  rather  than 
at  the  outer  surface  where  it  is  usually  taken,  provided  it  is 
desired  not  to  exceed  a  given  maximum  pressure  angle  on  the 
follower  pin. 

380.  WHEN  THE  PIN  is  MOVING,  THE  LINE  OF  CONTACT  between 
the  side  of  the  groove  and  the  side  of  a  cylindrical  pin  is  a  curved  line, 
and  one  phase  is  shown  in  end  projection  at  6^2  1 2  in  Fig.  149  and  in 
side  projection  at  Gi  Ii  in  Fig.  150.     When  the  pin  is  not  moving 
it  has  straight-line  contact  with  the  side  of  the  groove  as  shown 
at  GI  in  Fig.  149.     If  the  follower  pin  is  fixed  in  the  frame  that 
carries  it,  it  will  receive  wear  on  the  forward  stroke  entirely  within 
the  area  G  GG  IQ  I  G,  Fig.  149.     GQ  is  the  same  horizontal  distance 
from  the  vertical  centerline  through  A  as  (72  is  from  the  vertical 
centerline  through  A  2. 

381.  A  ROTATING   CYLINDRICAL  PIN   CANNOT   HAVE  PURE  ROLLING 

ACTION  AGAINST  THE  SIDE  OF  THE  GROOVE  in  a  cylindrical  cam,  for 
such  action  requires  at  least  that  two  rolling  curves  must  measure 
off  their  lengths  equally  on  each  other.  This  means  that  if  the  cir- 
cumference of  the  follower  pin  at  the  top  goes  a  certain  number  of 
times  in  the  curve  G  G^  G±,  the  circumference  at  the  bottom  must  go 
the  same  number  of  times  into  the  curve  /  /2  /4-  This,  of  course, 
cannot  happen  with  a  cylindrical  pin  for  the  circumferences  of  the 
pin  are  the  same  top  and  bottom  while  the  helical  curves  at  the  top 
and  bottom  of  the  groove  against  which  they  operate  are  totally 
different  in  length.  From  the  above  it  follows  that  there  will  be 
considerable  sliding  between  a  cylindrical  pin  and  cylindrical  cam 
and  that  this  will  be  greater,  the  greater  the  length  of  the  pin. 


190  CAMS 

382.  A  CONICAL  FOLLOWER  PIN  for  a  cylindrical  cam  is  shown  at 
MI  Ri  in  Fig.  150  and  in  end  view  in  Fig.  151.     In  the  latter  view  the 
line  G  Gi  is  tangent  to  the  helix  which  marks  the  center  of  the  top  of 
the  groove  and  A  2  Gi  is  normal  to  it,  giving  the  point  <72  at  which  the 
conical  pin  is  tangent  to  the  side  of  the  groove  at  the  outer  circum- 
ference.    The  conical  pin  is  here  taken  the  same  size  at  the  top  as  the 
cylindrical  pin  in  Fig.  149  and  consequently  the  line  A2  Gz  in  Fig.  151 
will  be  parallel  and  equal  to  the  line  A2  G%  in  Fig.  149.     Likewise 
/2,  Fig.  151,  is  the  point  of  tangency  at  the  inner  end  of  the  pin. 
These  points  of  tangency,  and  intermediate  ones,  will  determine  the 
line  of  contact  £2  /2  between  the  conical  pin  and  the  side  of  the 
groove  for  the  position  shown.     This  line  is  also  shown  in  side  pro- 
jection at  Gs  7s  in  Fig.  150.     If  the  pin  is  rigidly  attached  to  the  fol- 
lower framework  the  wear  on  the  pin  will  fall  on  the  area  represented 
by  the  surface  Si  83  Is  (73.     If  the  pin  is  free  to  turn  on  its  axis  it  will 
come  nearest  to  having  rolling  action  when  the  circumference  at  the 
bottom  of  the  conical  pin  is  to  the  circumference  at  the  top  as  the 
length  of  the  helix  B  B±  is  to  the  helix  A  A4  in  Fig.  149,  or,  when  the 
conical  vertex  of  the  roller  is  at  Oi,  Fig.  150,  on  the  centerline  of 
the  cylinder.     Conical  pins  give  thrust  in  an  axial  direction  and  conse- 
quently there  must  be  special  provision  in  the  follower  framework 
for  holding  the  pin  in  place.     Conical  pins  have  a  natural  advantage 
in  that  they  may  be  designed  to  move  in  axially  and  so  to  take  up 
wear  in  the  pin  and  in  the  groove. 

383.  AN  HYPERBOLOIDAL  FOLLOWER  PIN  is  shown  at  TI  Ui  Vi  Wi 
in  Fig.  150  and  in  end  view  in  Fig.  152.     In  the  latter  Figure  the 
lines  A  2  £2  and  A  2  1 2  are  perpendicular  to  the  top  and  bottom  helices 
of  the  groove  respectively,  the  same  as  the  similarly  lettered  lines  in 
Fig.  149.     If  the  diameters  TI  Wi  and  Ui  V\  are  taken  in  the  same 
ratio  to  each  other  as  the  lengths  of  the  top  and  bottom  helices  of  the 
groove  in  which  the  pin  rolls  the  closest  approximation  to  rolling 
action  will  be  obtained.     There  cannot  be  pure  rolling  of  the  hyper- 
boloidal  pin,  however,  on  the  side  of  the  groove,  for,  even  where  the 
circular  sections  of  an  hyperboloidal  pin  measure  themselves  off 
equally  on  the  corresponding,  helices  on  the  surface  against  which 
they  roll,  there  is  always  an  inherent  fundamental  endwise  or  longi- 
tudinal component  of  sliding  in  the  direction  of  the  axis  of  the 
pin  in  every  hyperloidal  action.     The  nature  and  amount  of  this 
characteristic  is  explained  in  some  of  the   books   on    descriptive 
geometry. 


MISCELLANEOUS   CAM   ACTIONS    AND    CONSTRUCTIONS      191 


192 


CAMS 


384.  To  LAY  OUT  THE  HYPERBOLOiDAL  PIN  the  lines  A2  (r2  and 
A 2  /2,  Fig  152,  and  the  circles  T  W  and  U  V  are  drawn.  The  straight 
line  from  G%  to  /2  is  drawn  and  used  as  the  generatrix  of  the  hyper- 
boloid,  the  outline  of  which  is  the  curved  line  T\  U\  in  Fig.  150. 


Points  on  this  curve  are  found  by  dividing  Gi  /2,  Fig.  152,  into  say 
four  equal  parts,  revolving  the  dividing  points  to  the  line  T  U  and 
then  projecting  them  to  the  equally  spaced  lines  which  are  drawn 
from  Si  to  £3  in  Fig.  150.  The  straight  line  G±  I±  will  be  the  line  of 


MISCELLANEOUS   CAM   ACTIONS    AND    CONSTRUCTIONS      193 

contact  between  the  pin  and  the  groove  at  the  maximum  pressure 
angle,  and  the  curved  hyperboloidal  line  Si  83  the  line  of  contact  at 
the  end  of  the  stroke.  The  wear  on  a  pin  fixed  in  the  follower  frame 
would  occur  on  the  hyperboloidal  surface  between  these  two  lines. 

385.  PLATES  FOR  CYLINDRICAL  CAMS.  Instead  of  actually  cutting 
grooves  in  cylinders,  flat  cam  plates  are  often  formed  and  then 
curved  to  the  diameter  of  a  cylinder  and  screwed  on.  The  follower 
pin  then  works  against  the  edge  of  the  formed  plate.  Such  a  case, 
illustrated  from  the  working  drawings  of  an  automatic  machine,  is 
given  in  Figs.  153  and  154.  Fig.  153  is  a  development  of  the  plate 
as  first  laid  out,  and  Fig.  154  an  end  view  of  the  plate  after  it  is 
curved  and  ready  to  be  applied  to  the  blank  cylinder  by  fastening 


FIG.   154. — CAM  PLATE  FORMED  TO  FIT  BLANK  CYLINDER 


screws  as  indicated.  The  developed  cam  surface,  as  shown  in  this 
illustration,  has  a  straight  line  and  gives  uniform  velocity  to  the 
follower  pin. 

386.  ADJUSTABLE  CYLINDRICAL  CAMS  IN  AUTOMATIC  WORK  FOR 
PROCESSES  INVOLVING  VARYING  SIZES  OF  PRODUCT.     By  making  a 
series  of  plates  of  varying  proportions,  but  curved  to  fit  a  single 
pitch  cylinder  which  remains  permanently  in  place,  and  by  fastening 
the  proper  series  of  plates  to  the  cylinder  for  a  given  job,  the  same 
automatic  machine  is  made  to  turn  out  products  of  varying  sizes 
without  removing  cam  bodies  from  shafts.     A  type  of  automatic 
machine  in  which  such  curved  cam  plates  are  used  is  the  screw  man- 
ufacturing machine.     A  diagram  of  a  blank  cylindrical  cam  with 
curved  cam  plates  fastened  on  is  shown  in  Fig.  11. 

387.  DOUBLE-SCREW  CYLINDRICAL  CAM.     This  is  the  cylindrical 
grooved  cam  with  specially  formed  follower  pin  or  head  adapted  to 
give  long  ranges  of  reciprocating  motion  to  a  follower  bar.     The 
cam  may  be  constructed  to  -give  uniform  or  variable  velocity  to  the 
follower  throughout  its  entire  range  of  motion;   also  to  give  periods 
of  rest  at  the  end  of  the  stroke,  if  desired,  ranging  anywhere  from  zero 


194 


CAMS 


to  periods  measured  by  the  time  required  for  the  cam  to  make  about 
!2/3  turns.  The  form  of  the  follower  pin  should  be  as  indicated  in 
general  way  in  Fig.  155  being  elongated  at  J,  Ji,  so  as  to  take  the 
open  space  at  two  intersecting  grooves  and  at  the  same  time  so  as 
not  to  bind  at  the  reverse  position  at  the  end  of  the  stroke  as  at  C, 
Fig.  155. 

388.  If  the  follower  motion  is  uniform  throughout  the  entire 
length  of  both  strokes,  the  right  and  left  grooves  C  D  M  and  L  F  C 
respectively  in  Fig.  155  will  have  a  true  helix  for  their  pitch  line. 
The  angle  at  the  end  of  the  screw  formed  by  the  intersecting  helices 
may  be  eased  off  as  shown  in  Fig.  157,  by  the  change  from  Ci  Y  to 
R  Y.  If  it  is  desired  to  have  the  follower  pause  at  the  end  of  the 
stroke,  the  helical  groove  will  run  into  a  groove  whose  sides  are 

K* — N  > QUTA 


~DT~P    E%3 


FIG.  155. — DOUBLE  SCBEW  CAM  WITH  HALF  TURN  STOP 

bounded  by  planes  perpendicular  to  the  axis  of  the  cylindrical  cam, 
as  shown  at  L  M,  Fig.  155.  The  length  of  the  stop  of  the  follower 
will  be  equal  to  the  time  required  for  the  cam  to  make  one-half  a 
turn  in  this  case,  as  shown  in  Fig.  155,  and  to  about  %  turn  if  made  as 
shown  at  C  D  E  F  in  Fig.  156.  This  is  about  the  practical  limit  for 
time  of  stop  for  a  simple  groove  construction. 

389.  PERIODS  OF  REST  CORRESPONDING  TO  MORE  THAN  ONE  REV- 
OLUTION OF  THE  CAM  may  be  obtained  by  a  special  attachment 
shown  at  H  N  in  Fig.  156.  A  movable  triangular  piece,  H ,  turns 
through  an  angle  of  a  few  degrees  on  a  pin  J  which  is  mounted  on  a 
small  sliding  block  that  is  constantly  pressed  to  the  left  by  the  spring 
N.  If  the  follower  head  V  K  is  imagined  to  move  around  the  cam,  it 
will,  shortly  after  passing  M  press  the  tip  S  of  the  piece  H  causing  it 
to  turn  slightly  on  the  pin  J  and  so  leaving  a  full  vertical  opening  at 
T  which  the  follower  head  will  enter  after  passing  L  on  its  way 
around.  The  follower  head,  having  entered  the  large  open  end  of 
the  groove  at  T  S  will  move  the  block  H  to  the  right  against  the 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS      195 

spring  pressure  at  N  as  it  passes;  and  as  the  follower  head  moves 
toward  and  past  S,  the  spring  N  will  cause  the  tip  T  of  the  block  to 
move  to  the  -left  and  so  open  full  the  inclined  groove  L  M  I  f or  the 
follower  head  to  enter  the  next  time  it  comes  around. 


K 


H 


FIG.  156. — DOUBLE  SCREW  CAM  WITH  f  AND  1 1  TURN  STOPS 
390.    A  SLOW  ADVANCE  OF  THE   FOLLOWER  AND  A  QUICK  RETURN 

may  be  obtained  with  the  double  screw  cam,  as  shown  in  Fig.  157, 
when  the  time  for  the  travel  of  the  follower  bar  P  K  Q  to  the  right  is 
represented  by  two  turns  of  the  cam,  while  the  return  motion  to  the 
left  will  be  made  in  one  turn  of  the  cam  as  shown  by  the  increased 
(double)  pitch  of  the  helical  groove  R  F  L.  To  accurately  represent 
the  helical  grooves  where  the  pitches  are  different,  as  in  this  case,  it 


FIG.  157. — DOUBLE  SCREW  CAM  FOR  SLOW  ADVANCE  AND  QUICK  RETURN  OF  FOLLOWER 

must  be  kept  in  mind  that  the  normal  distance  between  helices  must 
always  be  equal  to  the  effective  diameter  of  the  follower  pin  and 
that  this  will  give  a  wider  groove,  measured  parallel  to  the  axis,  when 
the  pitch  is  large  than  when  it  is  small.  This  is  shown  at  E,  Fig. 
157,  where  the  large  pitch  and  small  pitch  grooves  are  tangent  to 
the  circle  on  the  follower  head.  The  pitch  of  the  helix  or  groove  is 
the  longitudinal  or  axial  advance  of  the  follower  in  one  full  turn  of 
the  cylinder.  It  is  shown  in  Fig.  155  by  the  distance  D  0. 

391.  The  helical  groove  shown  in  Fig.  155  gives  uniform  motion 
to  the  follower,  and  the  helical  curves  which  bound  the  groove  are 


196 


CAMS 


constructed  in  the  same  way  as  described  in  Problem  15,  paragraph 
126  et  seq.  except  that  the  horizontal  spaces  A  H,  HI,  and  7  J, 
Fig.  57,  are  made  equal  for  the  helix  construction.  Any  practical 
variable  motion  may  be  obtained  with  this  type  of  cam  by  varying 
the  inclination  of  the  grooves,  the  smoothest  action  for  driving  the 
follower  from  one  end  of  the  stroke  to  the  other  regardless  of  inter- 
vening velocities  being  the  parabola  curve. 

392.  STRAIGHT  SLIDING  PLATE  CAMS.  The  sliding  plate  cam  M  N, 
Fig.  158,  is  but  a  simpler  form  of  the  rotating  cam.  The  figure  illus- 
trates two  common  uses  of  the  sliding  cam,  the  lifting  rod,  A  C,  on 
the  left  operating  a  poppet  valve,  and  the  bellcrank  B  E  F  on  the 
right  a  belt  shifter.  The  general  information  necessary  to  lay  out  this 


FIG.  158. — SLIDING  PLATE  CAM. 


cam  is  explained  in  detail  in  Problems  3  and  16  respectively.  In  the 
present  case  let  it  be  required  that  a  poppet  valve  be  lifted  the  dis- 
tance 2  units  with  uniform  acceleration  and  retardation  with  a  maxi- 
mum pressure  angle  of  30°;  find  the  length  and  form  of  cam  surface. 
The  cam  factor  for  the  parabola,  which  gives  uniform  acceleration 
and  retardation,  is  3.46  and,  therefore,  the  length  of  the  sliding  cam 
surface  will  be  2  X  3.46  =  6.92  as  shown  in  Fig.  158,  giving  a  pres- 
sure angle  of  30°  at  G.  The  curve  A  G  B  is  laid  out  in  the  regular 
way  for  the  parabola,  by  dividing  H  G  into,  say,  16  equal  parts  and 
taking  the  1st,  4th,  9th  and  16th  parts  and  drawing  horizontal  lines 
drawn  through  them  as  indicated.  K  G  is  divided  in  the  same  way. 
A  L  is  then  divided  into  8  equal  parts  and  vertical  lines  drawn  at 
each  of  the  points  to  meet  the  horizontal  lines  as  at  P,  Q,  etc. 

393.  The  bellcrank  B  E  F,  Fig.   158,  will  have  approximately 
uniform  angular  acceleration  and  retardation  if  the  cam  surface, 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS      197 

obtained  as  above  is  used,  but  if  exact  angular  acceleration  is  required 
the  method  described  in  paragraph  136  et  seq.,  for  the  cylindrical 
cam  chart  must  be  followed.  In  this  case  the  cam  chart  becomes 
the  sliding  plate  cam. 

394.  INVOLUTE  CAM.     The  involute  curve  may  be  used  for  cam 
outlines.     It  gives  a  characteristic  motion  almost  identical  with  the 
cam  having  a  straight-line  base  curve  (paragraph  32) ,  but  it  is  not  so 
simple  to  construct.     The  involute  cam  will  not  give  true  rnotion  to  a 
roller  follower  unless  the  ends  of  the  cam  working  surface  are  eased 
off,  as  they  are  in  the  straight-line  combination  cam,  or  the  logarith- 
mic combination  cam  (paragraphs  33,  56,  59,  and  199)  by  arcs  of 
circles  or  other  curves.     For  the  same  data  as  were  taken  for  com- 
parison of  cams  shown  in  Figs.  71,  75,  79,  etc.,  the  involute  curve 
gives  a  slightly  larger  cam  than  does  the  straight-line  base  curve, 
the  maximum  radius  being  2.78  for  the  former  and  2.65  for  the 
latter.     The  method  of  finding  the  maximum  radius  for  the  involute 
cam  will  be  explained  in  the  next  problem. 

395.  THE  INVOLUTE  is  POPULARLY  DEFINED  as  the  curve  that 
would  be  generated  by  a  point  on  a  string  which  is  being  unwound 
from  the  periphery  of  a  circular  disk,  the  string  always  being  kept 
taut  and  always  in  the  same  plane  as  the  disk. 

396.  THE  INVOLUTE  is  READILY  CONSTRUCTED,  according  to  the 
preceding  definition,  by  drawing  a  circle  of  any  size,  as  illustrated  at 
,-S  P  R,  Fig.  159;  taking  any  point  as  S  as  the  origin  of  the  involute; 
laying  off  a  series  of  equal  angles,  of  any  desired  unit,  as  at  S  0  M, 
M  ON,  etc. ;  drawing  tangents  to  the  circle  at  M,  N,  etc. ;  and  mak- 
ing the  lengths  M  Y,  N  U  of  the  tangents  equal,  successively,  to  the 
lengths  of  the  arcs  MS,  N  S,  etc.     This  latter  operation  may  be 
done  graphically  by  setting  the  small  dividers  to  a  step  of  ^  inch  or 
less,  starting  at  S  and  counting  the  steps  toward  M  until  M  is  reached 
or  passed,  and  then  counting  off  the  same  number  of  steps  in  the 
reverse  direction  going  along  the  tangent  line,  thus  obtaining  the  point 
Y  on  the  involute  curve.     This  graphical  method  of  stepping  off  dis- 
tances, although  generally  used,  is  apt  to  give  an  appreciable  cumu- 
lative error,  and  therefore  should  be  checked  by  a  simple  computa- 
tion as  follows:  Length  of  tangent  N  U,  for  example,  equals  length 
of  arc  NS,  equals  "S  X  2  X  3.14  X  angle  AT  OS  in  degrees     ^ 

360 

general  it  is  advisable  to  first  compute  and  draw  a  long  tangential 
line  as  R  W  at  180°  from  S,  and  then  if  six  equally  spaced  construe- 


198 


CAMS 


tion  points  are  used  as  at  M,  N,  etc.,  to  make  the  tangent  P  V  one- 
half  of  R  W]  the  tangent  at  M  Y,  one-third  of  P  F;  the  tangent  at 
N  U,  two-thirds  of  P  V,  etc. 

397.  PRESSURE  ANGLE  WITH  INVOLUTE  CAM.  Pressure  angle  is 
defined  as  the  angle  made  by  the  line  of  action  of  the  follower  and  the 
normal  to  the  pitch  curve  of  the  cam.  Therefore  if  the  follower  moves 
in  the  direction  0  V,  Fig.  159,  and  if  the  normal  to  the  involute  at  the 
point  V  is  V  H,  the  pressure  angle  is  H  V  K.  The  angle  H  V  K 
grows  smaller  as  the  point  V  is  moved  to  the  right  towards  W,  and 


FIQ.  159. — INVOLUTE  CURVE  AS  USED  SPECIFICALLY  IN  CAM  CONSTRUCTION 

larger  as  it  is  moved  to  the  left  towards  the  origin  of  the  curve  at  S. 
At  S  the  pressure  angle  would  be  90°  because  the  involute  is  tangent  to 
the  line  of  action  0  S  of  the  follower.  The  line  of  action  of  the  follower 
is  a  radial  line  in  the  type  of  cam  being  considered  in  this  problem. 
From  the  above  it  may  be  seen  that  there  are  a  series  of  points  on 
the  involute  where  there  are  definite  pressure  angles,  and  these  points 
will  be  noted  here  as  they  are  necessary  in  solving  a  specific  problem. 
398.  At  E,  Fig.  160,  the  pressure  angle  is  20°.  The  point  E  is 
obtained  by  laying  off  an  angle  of  88°  from  the  origin  of  the  curve,  as 
at  S  0  E.  The  other  points  for  pressure  angles  of  30°,  40°,  etc.,  are 
found  at  A,  D,  etc.,  by  using  the  values  given  in  the  following  table. 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS      199 

The  method  of  finding  these  values  will  be  given  in  a  later  paragraph 
for  those  who  may  be  interested. 


Pressure  angle 20° 

Initial  angle 88° 


30 


40° 

18° 


50° 

8° 


60° 
3° 


399.  PROBLEM  41.     REQUIRED  AN  INVOLUTE  CAM  that  will  move 
a  radial  follower  one  unit  while  the  cam  turns  60°  with  a  maximum 
pressure  of  30°. 

400.  To  solve  this  and  any  other  similar  involute  cam  problem 
it  is  necessary  to  construct  first  an  accurate  basic  involute  curve  of 
any  convenient  size  as  directed  in  the  preceding  paragraphs;    then 
to  lay  off  the  initial  angle  corresponding  to  the  given  pressure  angle 


•K 


FIG.  160. — SHOWING  THAT  ALL  SI~ES  OF  INVOLUTE  HAVE  THE  SAME  INITIAL  ANGLE  FOB 

EACH  PRESSURE  ANGLE 


in  the  problem.  In  this  problem  the  pressure  angle  is  given  as  30°, 
for  which  the  initial  angle  is  39j/£°  as  determined  from  the  table, 
and  this  latter  angle  is  laid  off  at  S  0  A  in  Fig.  159  where  the  basic 
curve  has  been  drawn.  From  0  A  lay  off  the  given  working  angle  of 
60°  as  at  A  0  F.  Draw  the  circular  arc  A  G  and  measure  the  dis- 
tance G  F.  Then  make  a  proportion  in  which  the  distance  G  F  is  to 
the  assigned  follower  motion  as  the  radius  0  A  is  to  the  desired  short- 
est radius  of  the  pitch  surface  of  'the  cam.  In  this  problem  G  F 
measures  1.12  units,  the  assigned  follower  motion  is  1  unit,  and  the 
radius  A  0  =  2.00 -units. 
Therefore, 


1.12  :   1.00  ::  2.00  :  x,    or    x  =  1.78, 


200 


CAMS 


equals  the  shortest  pitch  radius  of  the  cam.  With  this  value  known 
it  is  a  simple  matter  to  draw  the  required  involute  cam  as  in  Fig.  161, 
where  0  A  equals  x  as  just  computed,  H  A  K  equals  the  pressure 
angle  of  30°,  and  the  line  H  A  extended  is  tangent  to  the  base  circle 
E  Q,  which  is  necessary  in  the  drawing  of  the  desired  involute.  With 
these  items  laid  down,  the  involute  pitch  surface  from  A  to  F  is  con- 
structed in  detail  as  described  above  in  the  paragraph  devoted  to  the 
construction  of  the  involute,  and  briefly,  in  review,  as  indicated  in 
the  following  paragraph. 

401.  Lay  off  the  assigned  working  angle  of  60°  as  at  A  0  F, 
Fig,  161,     Make  0  F  equal  to  A  0  plus  the  assigned  motion  of  the 


Fia.  161. — PROBLEM  41,  INVOLUTE  CAM  FOR  RADIAL  FOLLOWER  BASED  ON  THE  SPECIFIC 

DATA 


follower  which  is  one  unit  in  this  problem.  Draw  F  Q  tangent  to  the 
base  circle.  H  T  is  tangent  to  the  base  circle.  Divide  the  arc  T  Q 
as  at  /  and  J  into  a  convenient  number  of  equal  parts,  three  being 
taken  in  this  illustration.  Draw  tangents  at  /  and  J.  Step  off  the 
distance  A  T I  on  I  h  thus  obtaining  /i,  etc.  Draw  the  involute 
curve  through  A,  7i,  Ji,  F,  thus*  obtaining  the  involute  pitch  surface 
of  the  cam.  The  working  surface,  if  a  roller  is  used,  is  found  by 
taking  the  radius  of  the  roller  and  drawing  a  series  of  arcs  with  /i,  Ji, 
etc.,  as  centers  and  drawing  a  curve  tangent  to  them  as  at  /2,  «/2, 
etc.  It  will  be  observed  that  a  working  curve  so  drawn  will  be  tan- 
gent to  the  last  construction  arc  at  Q2  on  one  side  of  the  cam  lobe  and 


MISCELLANEOUS   CAM    ACTIONS    AND    CONSTRUCTIONS      201 

tangent  at  $4  on  the  other  side,  giving  the  actual  tip  of  the  working 
cam  surface  at  Qs.  This  means  that  if  the  roller  follower  is  to  move 
with  a  velocity  characteristic  of  the  involute  curve,  that  its  ultimate 
stroke  will  be  less  than  the  desired  amount  by  the  distance  Qs,  Q*>. 
This  can  only  be  corrected,  when  a  roller  follower  is  used,  by  disre- 
garding the  involute  characteristics  at  the  end  of  the  stroke,  and  by 
arbitrarily  changing  the  true  working  surface  curve  from  «/2  to  §3 
so  that  the  curve  will  run  smoothly  from  /2  to  $5. 

402.  THE  INVOLUTE  CURVE  HAS  A  FIXED  INITIAL  ANGLE  FOR  EACH 
PRESSURE  ANGLE.     For  example,  the  initial  angle  SO  A,  Fig.  160, 
will  always  be  39^°  for  a  maximum  pressure  angle  of  30°  no  matter 
what  size  of  involute  is  used.     This  may  be  readily  shown  as  follows : 
Let  y  equal  the  angle  S  0  A  which  is  the  initial  angle  from  the  origin 
of  the  involute  to  the  point  where  the  pressure  angle  is  to  be  shown. 
Let  a  be  the  assigned  pressure   angle  as  represented  at  H  A  K. 
Then  the  angle  S  0  T  =  y  +  (90°  -  a°).     Let  x  equal  0  T,  the 
radius  of  the  base  circle  of  the  involute.    Then,  x  cot  a  =  A  T  =  arc 

y     I      QQ    n 

S  T  =  2  TT  x  - — — .     The  value  of  x  cancels,  and  for  a  pressure 

obu 

angle  of  30°  substituted  for  a,  y  is  found  to  be  39}^°.  Similarly  the 
initial  angle  is  found  to  be  18°  for  a  pressure  angle  of  40°. 

403.  INVOLUTE   SPECIALLY  ADAPTED    FOR  A  FLAT-SURFACE  FOL- 
LOWER.    The  involute  curve  is  naturally  adapted  for  an  oscillating 
cam  surface  where  a  flat-surface  follower  is  used,  and  in  this  case  it 
gives  a  uniform  linear  velocity  to  the  follower.     The  natural  advan- 
tage of  an  involute  cam  for  a  flat-surface  follower,  shown  in  Fig.  162, 
is  based  on  the  property  of  an  involute  that  the  tangent  X  Y  to  the 
base  circle  R  Q  is  normal  to  the  involute  as  at  F,  and  consequently 
that  the  perpendicular  line  at  T  Z  is  tangent  to  the  involute.    There- 
fore, T  U  may  represent  the  flat  surface  of  a  follower  collar  attached 
to  the  follower  rod  S  Si.     T\  U\  is  the  follower  in  its  highest  posi- 
tion, and  T2  C/2  in  its  lowest  position,  considering  that  only  the  part 
A  C  of  the  cam  surface  is  used.    An  involute  curve  cam  as  from  A  to 
C  would  always  be  tangent  to  the  flat  surface  of  the  follower  and  the 
line  of  contact  between  cam  and  collar  would  pass  through  the  center 
of  the  follower  rod,  moving  up  and  down  between  Y2  and  YI.     This 
means  that  there  is  no  pressure  angle  on  the  follower  rod  except  that 
due  to  friction.     This  last  feature  of  the  involute  cam  gives  it,  per- 
haps, its  greatest  practical  importance.     Where  it  is  desired  to  give 
the  follower  a  definite  velocity  and  acceleration  between  its  extreme 


202 


CAMS 


points  of  travel,  ^2,  Y\t  the  involute  cannot  be  used  and  the  method 
explained  in  paragraph  76  et  seq.  must  be  used. 

404.  OSCILLATING  POSITIVE  DRIVE  SINGLE-DISK  CAM.  This 
cam,  illustrated  in  Fig.  163,  might  be  compared  with  the  yoke  cam 
having  a  swinging  follower  instead  of  a  reciprocating  follower.  Its 
method  of  construction,  however,  differs  from  that  of  the  yoke  cam. 
In  the  illustration  the  oscillating  cam  L  KM  receives  its  motion 
through  the  link  F  (7,  the  point  F  swinging  through  the  arc  F\  FQ. 
The  follower  piece  CEDE  swings  about  the  fixed  center  B  through 
the  angle  E\  B  E&.  The  pitch  surfaces  CQ  Ci  and  D6  D\  are  found 
by  considering  the  cam  to  remain  stationary  while  the  follower 
revolves  around  it  in  such  a  way  as  to  retain  its  relative  working 


FIG.  162. — SHOWING  THAT  THE  INVOLUTE  is  SPECIALLY  ADAPTED  FOB  AN  OffsET  FLAT 

SURFACE  FOLLOWER 


position  at  all  phases.  The  detail  construction  necessary  to  do  this 
is  as  follows:  Through  the  point  B  draw  the  arc  B\  BQ  to  include  the 
same  angle  as  the  arc  F\  FQ.  Divide  arc  BI  BQ  into  a  number  of 
equal  parts  if  the  shaft  A  is  to  turn  with  uniform  angular  velocity; 
six  parts  are  used  here.  With  points  B\y  B%  as  centers  and  distance 
B  C  as  a  radius  draw  short  arcs  as  indicated  at  Ci,  €2,  etc.  The 
point  C  moves  in  an  arc  C'  C"  through  the  same  angle  as  does  the 
point  E. 

405.  If  it  is  desired  that  the  follower  move  with  angular  accelera- 
tion and  retardation  similar  to  that  produced  by  the  crank  curve, 
draw  the  semicircle  having  C  J,  Fig.  163,  for  a  radius,  divide  it  into 
the  same  number  of  parts  as  B\,  BQ  was  divided  (six  in  this  case)  and 
project  the  division  points  Ji,  J^  etc.,  to  the  arc  C'  C".  which  has  B 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS       203 

for  its  center.  Carry  these  last  points  around,  with  A  as  a  center, 
until  they  meet  the  corresponding  arcs  which  have  been  already 
drawn  with  B\t  BZI  etc.,  as  centers.  Thus,  the  points  C\  .  .  .  CQ 
of  the  pitch  surface  will  be  obtained.  The  points  D\  .  .  .  DQ 
of  the  companion  pitch  surface  are  obtained  in  the  same  way.  The 
radius  C  J  is  equal  to  one-half  the  chord  of  the  arc  Cr  C".  Taking 


a 


FIG.  163. — OSCILLATING  POSITIVE  DRIVE  SINGLE  DISK  CAM 

the  radius  of  the  roller  to  be  D  P  the  working  surfaces  KQ  M  and 
KI  L  are  obtained. 

406.  With  the  above  type  of  cam,  extreme  accuracy  is  necessary 
in  manufacture  to  overcome  any  binding  action  of  the  rollers  on  the 
cam  disk.  To  overcome  this  a  cam  construction  has  been  devised 
in  which  the  two  arms  B  C  and  B  Z),  Fig.  163,  are  entirely  separated, 
the  former  being  keyed  to  the  shaft  B  and  the  latter  free  to  turn  on 
shaft  B.  The  two  arms  are  then  connected  by  a  spring,  as  illustrated 
in  Fig.  166,  which  keeps  them  drawn  to  each  other,  and  both  having 


204 


CAMS 


the  desired  pressure  on  the  cam  surface.  To  prevent  too  great  a 
pressure  of  the  arms  on  the  cam  surface,  should  too  heavy  a  spring  be 
used,  a  stop  pin  is  cast  on  each  arm  and  these  stops  come  together 
just  as  the  follower  rollers  touch  the  cam  surface  when  newly  adjusted. 
407.  CAM  SHAFT  ACTING  AS  GUIDE.  A  special  form  of  construc- 
tion for  guiding  the  cam  follower  is  frequently  used  as  illustrated  in 
Figs.  164  and  165.  The  cam  B  in  Fig.  164  is  the  simple  radial  cam 
and  is  constructed  for  any  given  data  as  explained  in  paragraphs  49 
et  seq.  It  moves  the  roller  C,  which  is  attached  to  the  forked  arm 
R  D,  back  and  forth  in  approximately  a  radial  line  the  distance  A  M 
minus  A  L  which  is  equal  to  the  chord  of  the  arc  E  D.  The  arc  E  D 
measures  the  swing  of  the  shaft  F.  The  follower  rod  D  R  is  under 


FIG.  165. 


FIG.   164. — CAM  SHAFT  GUIDE  TAKES  PLACE  OF  CROSSHEAD  GUIDE 
FIG.   165. — POSITIVE  DRIVE  WITH  CAM  SHAFT  GUIDE 


definite  control  all  of  the  time,  although  its  form  of  construction  is 
extremely  simple  and  the  number  of  parts  a  minimum.  The  forked 
end  R  R  of  the  follower  rod  bears  with  a  snug  fit  against  the  two  sides 
of  the  cam  shaft,  or  against  adjustable  collars  attached  to  the  shaft. 
In  Fig.  164  the  follower  shaft  F  is  returned  to  its  initial  position  by 
means  of  the  spring  H. 

408.  POSITIVE  DRIVE  WITH  CAM  SHAFT  AS  GUIDE.  A  cam  giving 
positive  motion  where  the  cam  shaft  is  used  as  a  follower  guide  is 
illustrated  in  Fig.  165.  The  cam  itself  is  a  face  cam  and  is  con- 
structed for  any  given  data  as  directed  in  paragraphs  96  et  seq. 
The  pin  C  is  attached  to  the  follower  rod  D  R  and  is  moved  back  and 
forth  in  approximately  a  radial  position  by  the  amount  A  M  minus 
A  L,  equal  to  the  chord  of  D  E.  The  forked  end  of  the  follower  rod, 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS       205 

bearing  against  the  sides  of  the  cam  shaft  A,  together  with  the  guided 
end  D  of  the  rod  give  it  a  motion  that  is  under  control  at  all  phases, 
and  this  with  a  minimum  amount  of  mechanical  construction.  The 
shaft  F  is  under  positive  cam  control  all  the  time  on  account  of  the 
use  of  the  face  cam,  and  no  return  spring  is  necessary  as  in  Fig.  164. 

409.  POSITIVE  DRIVE  DOUBLE  DISK  RADIAL  CAM  WITH  SWINGING 
FOLLOWER.     A  special  form  of  cam  and  follower  construction,  where 
positive  action  is  desired,  is  shown  in  Fig.  166  where  the  following 
data  are  so  taken: 

(a)  That  two  follower  arms  10  units  long  shall  each  swing  through 
an  angle  of  20°  with  uniform  acceleration  and  retardation    while 
two  corresponding  radial  cams  turn  through  135°,  the  drive  to  be 
positive  with  each  roller  having  a  single  point  of  contact. 

(b)  That  the  follower  arms  shall  be  returned  with  positive  action 
while  the  cams  turn  through  225°. 

(c)  That  the  angle  between  the  two  radial  follower  arms  shall 
be  50°. 

410.  The  follower  shaft  A,  Fig.  166,  is  first  laid  down,  the  angle 
of  follower-arm  swing  of  20°  then  drawn  as  at  B  A  C,  and  finally  the 
10  units  for  follower-arm  length  laid  off  at  the  initial  position  A  B. 
The  horizontal  centerline  E  0  for  the  cam  shaft  is  then  drawn  across 
the  arc  B  C  so  that  the  midpoint  D  is  as  much  above  it  as  the  end 
points  B  and  C  are  below.     The  radius  0  D  of  the  pitch  circle  is  com- 
puted in  the  usual  way,  taking  the  chord  B  C  for  the  distance  moved 

. ,     .  „                        ™        n  n       B  C  X  3.46  X  360 
by  the  follower  point.     Then  D  0  = =  5.16,  thus 

O.Zo  X  loO 

locating  the  cam  center  0.  The  circle  represented  by  A  AI  is  then 
drawn  and  the  pitch  surface,  indicated  by  the  short  portion  B  BI 
is  constructed  in  exactly  the  same  manner  as  explained  in  Problem  8. 
The  size  of  the  roller  is  assumed  as  shown  at  B  F  and  the  working 
surface  of  the  operating  cam  F  G  is  drawn.  The  operating  arm  A  B 
is  keyed  to  the  shaft  A . 

411.  The  return  arm  A  H,  Fig.  166,  is  not  keyed  to  the  shaft  A 
but  turns  freely  on  it  instead.     The  motion  of  this  arm  should  be 
identical  with  that  of  the  arm  A  B  and,  therefore,  the  swinging  arc 
H  J  of  the  center  of  the  follower  roller  is  made  the  same  as  the  arc 
B  C,  and  it  is  similarly  divided.     The  pitch  surface  of  the  return 
cam  is  represented  in  part  at  H  HI  and  is  found  in  exactly  the  same 
way  as  directed  in  the  preceding  paragraph,  and  the  working  surface 


206 


CAMS 


K  L  drawn.  The  spring  at  M  exerts  more  pull  than  is  required  to 
return  the  follower,  and,  therefore,  it  holds  the  two  follower  arms 
against  the  cams  with  practically  uniform  pressure,  even  should 
there  be  slight  inaccuracies  in  workmanship,  or  wear  in  the  contact 
surfaces.  A  lug  is  attached  to  each  follower  arm  as  shown  at  P  and  Q, 


FIG.   166. — OSCILLATING  POSITIVE   DRIVE  DOUBLE   DISK  CAM 

and  these  act  as  stops  in  preventing  excessive  pressure  of  the  rollers 
on  the  cam  surfaces.  Cams  used  in  this  way  have  been  called  duplex 
cams. 

412.  ROTARY  SLIDING  YOKE  CAMS  GIVING  INTERMITTENT  HAR- 
MONIC MOTION.  A  yoke  cam  driven  by  sliding  contact  instead  of 
roller  contact  is  shown  in  Fig.  167.  The  cam,  in  this  figure,  is  in  the 
form  of  an  equilateral  triangle  bounded  by  equal  circular  arcs  having 
a  radius  equal  to  the  straight  sides  of  the  inscribed  triangle.  The  cen- 


MISCELLANEOUS   CAM   ACTIONS   AND    CONSTRUCTIONS       207 

ters  for  the  circular  arcs  are  at  the  apexes  of  the  triangle.  One  of  the 
apexes  of  the  cam  is  at  the  center  of  the  driving  shaft.  The  motion 
given  to  the  follower  yoke  will  be  an  intermittent  one,  dwelling  at 
the  ends  of  the  stroke,  and  the  total  travel  will  be  equal  to  the  radius 
of  the  cam  surface.  The  follower  will  travel  from  one  end  of  its 
stroke  to  the  other  with  a  simple  harmonic  motion  the  same  as  with 
crank  and  connecting  rod  where  the  connecting  rod  is  assumed 
to  be  of  infinite  length.  Or,  the  motion  during  the  stroke  will  be  the 
same  as  with  the  Scotch  yoke,  or  crank  and  slotted  crosshead,  where 
the  radius  of  the  crank  is  one-half  the  radius  of  the  present  cam  sur- 
face. 

413.  A  diagram  of  the  motion  of  the  yoke  follower  in  Fig.  167  is 
shown  at  M  N  S.     With  the  cam  turning  as  shown  by  the  arrow,  the 


Fro.  167. — SLIDING  YOKE  CAM  GIVING  HARMONIC  MOTION 

follower  H*  K  will  move  the  distance  M  N  while  C  on  the  cam  turns 
60°  to  D,  and  the  cam  edge  at  C  will  do  the  driving  with  a  scraping, 
sliding  action.  While  C  is  turning  60°  from  D  to  E,  the  follower  will 
remain  at  rest;  while  C  is  turning  60°  from  E  to  F  the  curved  surface 
A  C  of  the  cam  will  be  driving  the  follower  the  distance  0  P  with  a 
rubbing,  sliding  action  and  increasing  velocity;  while  C  is  turning 
from  F  to  G  the  cam  edge  C  will  again  be  driving,  the  follower  moving 
the  distance  P  Q  with  a  scraping,  sliding  action  and  decreasing  velocity. 
The  smooth  working  surface  of  the  follower  yoke  is  shown  from  F  to  B, 
while  the  recessed  surface  as  at  W  may  be  left  rough  cast.  The 
velocity  and  acceleration  diagrams  for  the  equilateral  sliding  yoke 
cam  here  described  have  the  same  characteristics  as  those  shown  for 
the  ordinary  crank  curve  cam,  illustrated  in  Figs.  88  and  89. 

414.  ROTARY  SLIDING  YOKE  CAM  GIVING  RECIPROCATING  HAR- 
MONIC MOTION.     A  circle  passing  through  the  points  ABC,  Fig.  167 


208 


CAMS 


FIG.   168. — SLIDING  YOKE  CAM 
GENERAL  CASE 


would  represent  the  surface  of  a  cam  attached  to  the  crank  A  J,  and 
such  a  cam,  instead  of  the  equilateral  curved  side  cam,  which  is  shown, 
would  give  a  simple  harmonic  motion  to  the  follower  yoke  without 
finite  periods  of  rest  at  the  ends  of  the  stroke.  Such  a  circular  cam 
would  be  an  equivalent  of  a  crank  and  slotted  crosshead  where  the 
radius  of  the  crank  would  be  equal  to  the  radius  of  the  cam  circle. 

415.  A  ROTARY  SLIDING  YOKE  CAM,  GENERAL  CASE,  with  the  Cam 

surface  entirely  surrounding  the  shaft  is  shown  in  Fig.  168.     To  lay 

out  this  cam  for  a  definite  range  of 
motion,  say  2  units,  draw  the  indefinite 
circular  arc  B  C  with  any  desired  radius 
and  the  arc  E  D  with  the  same  center 
and  with  a  radius  2  units  larger.  Then 
with  a  radius  equal  to  A  B  plus  A  E  and 
a  center  anywhere  on  the  arc  E  D  draw 
the  arc  C  D  until  it  intersects  E  D  as  at 
D.  With  D  as  a  center  and  the  same 
radius  as  before  draw  the  arc  E  B  com- 
pleting the  cam.  The  student  should 

be  able  to  determine  the  angles  traveled  by  the  cam  while  the 
follower  is  at  rest,  the  angles  of  motion,  the  range  of  motion  of  the 
follower,  and  the  exact  portion  of  the  follower  working  surface  which 
has  to  resist  the  wear  due  to  sliding  action. 

416.  CAM  SURFACE  ON  RECIPROCATING  FOLLOWER  ROD.     In  some 
special  forms  of  cam  construction  it  is  more  convenient  to  place  the 
cam  curve  on  the  follower  than  on  the  driver.     Such  a  case  is  illus- 
trated in  Fig.  169  where  the  cam  curve  E  F  E'  is  on  the  sliding  fol- 
lower bar  K  G.     The  driving  crank  A  F  carries  a  pin  at  F  which  slides 
in  the  cam  groove.     The  mechanism  here  shown  is  a  modification 
of  the  Scotch  yoke,  or  "infinite  connecting  rod."     The  motion  in 
this  case  is  such  that  the  follower  remains  stationary,  while  the 
driving  shaft  turns  through  the  angle  C'  A  C.     The  curve  C'  F  C  is 
an  arc  of  a  circle  with  A  as  a  center.     The  follower  then  picks  up 
motion  comparatively  slowly,  the  point  G  being  at  the  points  1,2,  8, 
etc.,  when  the  crank  pin  F  is  at  the  points  which  are  correspondingly 
represented  in  Roman  numbers.     When  the  crank  pin  is  at  J,  G  is  at 
N  and  it  then  moves  very  rapidly  from  N  to  P  while  the  crank  pin 
travels  from  J  to  Q.     Very  often,  in  cam  work,  the  driving  shaft  A 
has  only  an  oscillating  motion  through  90°  or  less.     If  the  curve 
C'  F  C  is  changed  slightly  so  as  not  to  be  an  arc  of  a  circle  with  A 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTION       209 


as  a  center,  the  end  G  of  the  follower  bar  will  not  come  to  rest  for  a 
definite  period  at  the  end  of  the  stroke,  but  it  will  have  a  slow,  power- 
ful motion  which  may  be  made  use  of  in  manufacturing  processes 
where  compression  is  required. 

417.  PROBLEM  42.  DEFINITE  MOTION  WHERE  CAM  SURFACE  is 
ON  FOLLOWER  ROD.  In  Fig.  169  a  follower  rod  G  K  has  a  cam  surface 
formed  at  the  left-hand  end  from  E  to  E',  and  it  is  driven  by  a  simple 
crank  pin  represented  at  F  so  as  to  secure  a  desired  or  known  motion. 
In  the  illustration  let  it  be  desired: 

1st.  That  the  follower  rod  shall  remain  at  rest  at  the  head  end  of 
the  stroke  while  the  driving  crank  pin  turns  45°  (22J^°  on  each  side 
of  the  centerline  A  F). 


X* 

H 

-F 

P6       5       4 

N        2       1 

1  -'       '       4' 

3'         2'     1' 

*x. 

FIG.  169. — CAM  SURFACE  ON  RECIPROCATING  FOLLOWER  ROD 


2d.  That  the  follower  will  be  moved  to  the  left  a  distance  G  N 
with  uniform  acceleration  while  the  crank  pins  turns  67  J^. 

3d.  That  the  follower  shall  move  the  remainder  of  the  stroke 
from  N  to  P  while  the  crank  pin  turns  90°. 

4th.  That  the  follower  rod  shall  move  in  reverse  order  on  the 
return  stroke  from  P  to  G. 

418.  Before  starting  the  solution  of  this  problem  it  should  be 
stated  that,  one  cannot,  because  of  either  theoretical  or  practical 
considerations,  or  both  combined,  always  secure  desired  results  in 
cams  of  this  type  where  arbitrary  distance  and  motion  assignments 
are  given  as  in  this  illustration.  It  is  nevertheless  advisable  to 
complete  the  solution  of  the  problem,  if  possible,  on  the  basis  of  the 
desired  data,  because  one  can  then  make  the  necessary  modifications 


210  CAMS 

with  a  sure  knowledge  that  the  least  departure  has  been  made  from 
the  theoretical  or  assigned  conditions. 

419.  The  method  of  solution  for  the  above  data  is  as  follows: 
Assume  the  driving  crank  length  A  F  and  draw  the  crank-pin  circle 
F  J  M .     Lay  off  the  angle  F  AC  equal  to  22^°.     The  circular  arc 
F  C  will  then  be  part  of  the  pitch  line  of  the  follower  cam  head,  and 
while  the  crank  pin  F  is  moving  through  this  arc  the  follower  rod  will 
not  move  at  all.     To  secure  uniform  acceleration  of  the  follower  for 
the  distance  G  N,  divide  G  N  into  9  equal  parts  and  mark  the  1st, 
and  4th  division  points  as  indicated  at  1'  and  2'  in  the  figure.     This 
will  be  the  first  step  in  securing  the  uniform  acceleration  called  for 
because  the  distance  from  G  to  1'  will  be  one  unit,  from  1'  to  2'  will  be 
three  units,  and  from  2'  to  3r  will  be  five  units.     By  dividing  G  N  into 
three  parts  as  here  described,  three  construction  points  will  be  secured 
on  the  cam  curve.     If  more   construction  points  are  desired,  GN 
may  be  divided  in  16  equal  parts  and  the  1st,  4th  and  9th  inter- 
mediate  division   points   taken,    thus    obtaining   four   construction 
points  on  the  part  of  the  pitch  surface  of  the  cam  from  C  to  E.     Like- 
wise, if  five  construction  points  are  desired,  GN  would  be  divided 
into  25  equal  parts,  and  the  1st,  4th,  9th,  and  16th  division  points 
taken. 

420.  Since  the  motion  from  G  to  N  is  to  take  place  while  the  crank 
pin  moves  67J^°  as  called  for  in  the  data,  and  since  three  construction 
points  have  been  used  in  this  illustration,  the  67J/2°  arc  from  C  to  J 
is  now  divided  into  3  equal  parts  as  indicated  at  7  and  II  in  Fig.  169. 
At  /  draw  a  horizontal  line  and  make  the  distance  I-R  equal  to  I'-G; 
at  II  make  the  distance  II-S  equal  to  2'-G]   and  at  J,  make  the  dis- 
tance III-E  equal  to  3'-G.     A  curve  through  the  points  C,  R,  S 
and  E  will  be  the  pitch  line  for  the  cam  surface  on  the  follower  rod 
for  uniform  acceleration  from  G  to  N.     The  p  oint  3'  coincides  with  N. 

421.  The  part  of  this  pitch  curve  from  R  to  E  is  shown  by  a  dash 
line  and  is  not  practical  because  of  the  sharp  curvature  from  S  to  E, 
which  would  produce  too  large  a  pressure  angle  and  this  in  turn 
would  give  a  large  bending  moment  on  the  follower  arm  and  large 
side  pressure  in  the  bearing  H.     This  part  of  the   curve  should, 
therefore,  be  modified,  and  a  good  plan  on  which  to  effect  the  mod- 
ification is  to  start  by  making  the  pressure  angle  as  large  as  is  prac- 
tically allowable  and  then  to  keep  the  new  curve  as  near  to  the  old  as 
possible.     A  maximum  pressure  angle  that  is  safe  under  all  ordinary 
circumstances  is  30°  and,  therefore,  the  first  step  in  the  modification 


MISCELLANEOUS    CAM   ACTIONS   AND    CONSTRUCTIONS      211 

will  be  to  draw  a  vertical  line  through  E,  the  end  of  the  theoretical 
curve,  and  make  an  angle  of  W  E  V  equal  to  the  maximum  practical 
pressure  angle  of  30°.  The  line  V  E  is  then  produced  until  it  crosses 
the  dash  curve,  and  a  smooth  curve  is  next  drawn  so  as  to  connect  the 
straight  line  and  the  original  curve.  This  will  leave,  in  this  case, 
E  T  as  a  straight  30°  line,  T  R  as  a  new  assumed  part  of  the  pitch 
curve,  and  R  F  as  the  portion  of  the  original  curve  that  remains.  If 
the  cam  is  to  turn  slowly,  or  if  the  load  on  the  cam  is  not  large,  a 
greater  pressure  angle  could  be  taken  at  W  E  V  and  then  the  arbi- 
trary new  curve  would  come  closer  to  the  original  or  theoretical 
curve. 

422.  The  practical  pitch  line  of  the  cam  is  now  found  to  be 
F  C  R  T  E.     The  cam  will  run  smoothly  and  the  variation  in  the 
motion  of  the  cam  from  the  originally  desired  motion  may  be  par- 
tially indicated  by  pointing  out  that  the  end  G  of  the  follower  will  be 
at  2,  and  at  4  instead  of  2'  and  4'  as  originally  intended.     This  varia- 
tion may  be  most  completely  shown  by  a  velocity  diagram  which  will 
be  taken  up  in  a  succeeding  paragraph. 

423.  The  pitch  curve  F  T  E,  it  will  be  noted,  has  been  con- 
structed to  give  a  definite  practical  action  to  the  follower  from  G  to  N. 
Since  the  curve  F  E  is  now  determined,  and  since  the  crank  pin  must 
drive  through  the  same  cam  slot  from  E  to  F  while  it  turns  through 
the  remaining  arc  J  Q,  it  follows  that  the  motion  of  the  rod  from 
N  to  P  cannot  be  assigned,  and  that  it  must  be  taken  as  it  comes. 
To  find  out  in  a  general  way  what  this  motion  will  be  it  is  only 
necessary  to  pursue  in  reverse    order  the   methods  already  used; 
i.e.,  to  lay  off  the  distance  T-IV  at  G-4,  the  distance  R-V  at  G-6, 
etc.     By  noting  the  distances  N~4,  4~5,  etc.,  which  the  follower  rod 
travels  in  uniform  periods  of  time,  some  useful  idea  of  the  retardation, 
and  consequently  of  the  smoothness  of  action  of  the  cam  may  be 
obtained  as  it  approaches  the  inward  end  of  its  stroke.     In  the 
illustration  the  follower  rod  will  slow  down  perceptibly  from  N  to  4, 
and  have  slightly  higher  but  a  fairly  uniform  velocity  from  4  to  5, 
and  from  5  to  6.     It  will  retard  rapidly  from  6  to  the  end  of  the 
stroke. 

424.  The  lower  part  of  the  pitch  curve  from  F  to  E'  will  be  made 
symmetrical  with  the  upper  part  from  F  to  E  in  this  problem  thus 
making  the  action  of  the  follower  on  the  return  stroke  the  reverse 
of  what  it  is  on  the  forward  stroke.     If  it  were  desired,  the  curve 
F  E'  could  be  constructed,  by  the  methods  described  above  to  give 


212 


CAMS 


the  same  characteristic  motion  to  the  follower  on  the  return  stroke 
as  it  did  on  the  forward  stroke. 

425.    AN    EXACT    KNOWLEDGE    OF    THE    EFFECT    OF    ARBITRARILY 

CHANGING  THE  THEORETICAL  CURVE  R  S  E,  Fig.  169  to  R  T  E  may 


? 

r 

i 

> 

H 

>F 

PCy       5       4      N        21 

4'     3'         2'     i 

\ 

FIQ.  169. — (Duplicate)  CAM  SURFACE  ON  RECIPROCATING  FOLLOWER  ROD 


be  readily  obtained  by  a  time-velocity  diagram  construction  as 
illustrated  in  Fig.  170.  In  the  latter  figure  let  the  length  of  the  base 
line  F  Q  represent  the  time  necessary  for  the  crank  pin  to  make  a  half 
revolution  from  F  to  Q,  Fig.  169.  Since  the  crank  pin  is  assumed 

to  travel  with  uniform  velocity,  the  line 
F  Q,  Fig.  170,  is  divided  into  eight  equal 
parts  the  same  as  is  the  semi-circle  F  Q  in 
Fig.  169.  The  velocity  of  the  follower 
at  each  of  the  construction  points  is 
then  found  as  indicated  in  the  following 
paragraph.  . 

426.  At  the  point  //,  for  example,  in 
Fig.  169,  draw  the  tangential  line  II-B 
of  any  desired  length.  This  line  will 
represent  the  velocity  of  the  crank  pin 
in  feet  per  second,  which  may  be  readily 
computed,  for,  if  the  crank  A  F  is  4  inches  long  and  makes  120  revo- 
lutions per  minute  the  point  F  will  be  moving  with  a  velocity  of 

^  X  2  X  3.14  X™  =  4.19  feet  per  second. 


Q    6     5 


1    C  F 


Fio.  170. — PROBLEM  42,  TIME- 
VELOCITY  DIAGRAM  FOR  RE- 
CIPROCATING FOLLOWER  ROD 
SHOWN  IN  FIQ.  169 


MISCELLANEOUS   CAM    ACTIONS   AND    CONSTRUCTIONS      213 

Through  the  point  B  draw  a  line  B  D  parallel  to  the  line  that  is  tan- 
gent to  the  cam  pitch  curve  at  T.  The  line  V  E,  continued,  is  tan- 
gent to  the  cam  curve  at  T  because  it  will  be  remembered  that  the 
practical  curve  from  Rio  T  was  taken  so  as  to  be  tangent  at  its  upper 
end  to  the  straight  line  E  T.  The  distance  II-D  will  represent  the 
velocity  in  feet  per  second  with  which  the  follower  rod  is  sliding 
through  the  bearing  at  H.  This  velocity  is  laid  off  in  the  time-velocity 
diagram  in  Fig.  170  at  2-D.  In  a  similar  manner  other  points  on  the 
solid-line  curve  C  D  Q  may  be  found.  This  curve  shows  at  a  glance 
just  how  fast  the  cam  follower  is  moving,  at  every  phase  of  its  stroke. 

427.  The  dash  line  construction  in  Fig.  170  shows  the  follower 
velocities  called  for  in  the  original  data,  but  abandoned,  as  explained 
above,  because  of  the  large  pressure  angle  involved.     The  point  0 
on  the  dash  curve  is  found  by  drawing  the  line  B  0,  Fig.  169,  through 
B  parallel  to  the  short  straight  dash  line  which  is  shown  tangent  to  the 
theoretical  curve  at  S.     Then  II-O  would  represent  the  velocity  of 
the  follower  bar  at  phase  II  if  the  original  data  were  used.    As  a 
check  on  the  accuracy  of  the  construction  the  points  C,  L,  0  and  X, 
Fig.  170,  should  all  be  on  a  straight  inclined  line,  because  C  X  is  a 
velocity  line  and  it  must  show  uniformly  increasing  velocity  for  the 
follower  in  order  that  there  may  be  uniform  acceleration  as  called 
for  in  the  original  data. 

428.  The  difference  between  the  solid  and  dotted  parts  of  the 
velocity  diagram  in  Fig.  170  shows  the  effect  on  the  velocity  of  the 
follower  of  arbitrarily  changing  the  theoretical  cam  curve  R  S  E, 
Fig.  169,  to  the  more  practical  cam  curve  R  T  E. 

429.  PROBLEM  43.  CAM  SURFACE  ON  SWINGING  FOLLOWER  ARM. 
When  the  cam  surface  is  on  the  follower  and  it  is  desired  that  the 
follower  shall  have  a  swinging  motion  instead  of  a  rectilinear  recip- 
rocating motion  as  it  had  in  Fig.  169,  the  method  of  construction 
will  vary  in  detail  as  illustrated  in  Fig.  171.     The  data  for  Fig.  171 
are,  that  the  driving  crank  A  C  with  a  crank-pin  roller  at  G  shall 
swing  the  follower  shaft  B  through  an  angle  of  30°  counterclockwise 
with  uniformly  increasing  and  decreasing  angular  velocity  while  the 
driving  shaft  turns  through  60°  with  uniform  angular  velocity  in  the 
same  direction. 

430.  The  method  of  locating  points  on  the  curve  C  F  of  the  fol- 
lower cam  pitch  surface,  Fig.  171,  follows:   Divide  the  assigned  30° 
arc,  C  E,  into  any  number  of  parts,  say  six,  which  are  as  to  each  other 
as  1,  3,  5,  5,  3,  1.     This  will  provide  for  the  uniformly  increasing  and 


214 


CAMS 


decreasing  motion  to  the  shaft  B.  Divide  the  assigned  60°  driver 
arc;  C  D,  into  six  equal  parts.  The  method  of  locating  the  point  L, 
which  is  the  second  construction  point  on  the  cam  curve,  will  be  taken 
for  explanation  purposes.  Other  points  are  found  in  the  same  way. 
Draw  a  radial  line  B  2  through  the  second  construction  point,  con- 
tinuing it  to  J  which  is  on  an  arc  which  passes  through  77  on  the  arc 
C  D .  Lay  off  the  arc  J  K  at  II-L  thus  obtaining  the  point  L  on 
the  cam  curve.  This  form  of  cam  has  positive  action.  When  it  is 


Fia.  171. — PROBLEM  43,  CAM  SURFACE  ON  SWINGING  FOLLOWER  ARM  HAVING  UNIFORIL 
ANGULAR  ACCELERATION  AND   RETARDATION 

allowed  to  reach  a  dead  center  position  as  shown  in  Fig.  171,  auxiliary 
action  will  be  required  in  starting. 

431.  EFFECT  OF  SWINGING  TRANSMITTER  ARM  BETWEEN  ORDINARY 
RADIAL  CAM  AND  FOLLOWER.     In  Fig.  172  let  B  C  D  E  F  be  an  ordi- 
nary radial  cam  with  straight  sides  as  at  B  H  rounded  off  by  circular 
arcs  with  center  as  at  G.     Let  /  J  K  be  the  swinging  transmitter  arm 
with  the  working  surfaces  at  J  and  K  as  arcs  of  circles  with  centers 
at  L  and  M  respectively.     Let  N  Nf  be  the  centerline  of  the  follower 
rod  which  moves  straight  up  and  down. 

432.  In  order  to  reach  a  useful  understanding  of  the  action  of  this 
type  of  cam  construction  it  will  be  necessary  to  learn  the  rate  of 


MISCELLANEOUS    CAM    ACTIONS   AND    CONSTRUCTIONS      215 

change  of  velocities  in  the  follower  parts  so  as  to  judge  the  accelera- 
tions and  retardations  which  cause  the  most  trouble  at  high  speeds, 
also  to  learn  the  rates  of  sliding  at  J  and  K,  and  then  to  balance  these 
against  the  pressure  angle  produced  by  the  same  radial  cam  with  an 
ordinary  direct  roller-end  follower. 


Fio.  172. — SWINGING  TRANSMITTER  ARM  WITH  SLIDING  ACTION 

433.  The  method  of  analyzing  the  cam  action  in  Fig.  172  will  be 
pointed  out  by  using  six  equally  spaced  construction  points  during 
the  period  that  the  surface  B  C  is  in  action,  the  cam  turning  as  shown 
by  the  arrow.  To  obtain  the  positions  of  the  six  points  for  analysis 
one  cannot  divide  the  subtended  arc  B  P  of  the  working  surface  arc 
B  C  into  six  equal  parts  where  a  swinging  follower  arm  is  used,  as  may 
be  recalled  from  Problems  8  and  11.  Instead,  it  is  convenient  foi 
analytical  construction  purposes  to  revolve  the  swinging  follower 
arm  around  the  cam  with  uniform  angular  velocity  while  the  cam 


216  CAMS 

remains  stationary.  The  detail  work  necessary  to  accomplish  this 
is  done  first  by  drawing  an  arc  of  a  circle  through  /  with  A  as  a  center, 
finding  where  /  is  on  this  arc  at  the  beginning  and  end  of  action  while 
the  arm  I  J  slides  on  B  C,  and  then  dividing  the  arc  of  swing  of  /  into 
six  equal  construction  parts. 

434.  The  initial  position  of  7,  Fig.  172,  is  found  by  laying  off  the 
distance  L  /  from  the  point  B  on  the  radial  line  A  B,  thus  obtaining 
the  point  0;  then  using  0  as  a  center  and  a  radius  equal  to  L  I  draw 
a  new  arc  to  intersect  the  arc  through  /  at  /i.    This  will  be  the  posi- 
tion of  7  when  the  swinging  arm  is  tangent  to  the  cam  at  B\  in  a  sim- 
ilar manner  IQ  will  be  found  to  be  the  position  when  the  arm  is  tan- 
gent at  C.     With  the  arc  I\  IQ  obtained  and  divided  into  six  equal 
parts,  it  is  no  longer  necessary  or  convenient  to  consider  the  center  / 
as  revolving  about  A,  and  it  will,  therefore,  be  considered  as  fixed 
in  further  work,  the  next  step  of  which  will  be  to  find  the  six  corre- 
sponding positions  of  the  point  L.     This  is  readily  done  by  drawing 
an  arc  through  L  with  7  as  a  center  and  then  taking  7  L  as  a  radius 
and  the  point  /4,  for  example,  as  a  center  and  drawing  an  arc  such  as 
one  of  the  short  ones  shown  at  0±.     Then  with  a  radius  equal  to  L  J, 
find  by  trial,  a  point  on  the  arc  just  drawn  which  will  be  a  center  for 
an  arc  that  is  tangent  to  -B  C  of  the  cam.     This  center  is  shown  at  0* 
and  the  tangent  arc  is  shown  at  B±.    With  A  as  a  center  draw  an 
arc  through  the  point  0±  until  it  cuts  the  arc  through  L  already 
drawn,  as  at  Z/4.     In  the  same  manner  the  six  points  on  the  arc 
through  L  are  found,  and  the  corresponding  points  of  tangency  on 
the  cam  outline  B  C  are  obtained  as  shown  from  B  to  (7. 

435.  THE  LOCUS  OF  THE  POINT  OF  CONTACT  may  now  be  found,  as 
at  R  J  RQ,  Fig.  172,  as  follows :   To  find,  for  example,  the  point  R±, 
draw  two  intersecting  arcs,  one  having  L  J  for  a  radius  and  L±  for  a 
center  and  the  other  having  A  B±  for  a  radius  and  A  for  a  center. 
Similarly  other  points  on  R  J  RQ  are  found. 

436.  THE  ANGULAR  VELOCITY  CURVE  FOR  THE  SWINGING  FOLLOWER 
ARM  may  now  be  readily  found  and  its  acceleration  and  retardation 
judged.     Let  S  T  represent  the  linear  velocity  of  a  point  S  at  radius 
A  S  on  the  cam.     Then  a  point  at  B±  on  the  working  surface  of  the 
cam  has  a  linear  velocity  of  S'  T'  and  this  value  is  laid  off  at  R±  TV 
where  the  point  B±  is  in  action.     The  component  of  R±  T<2.  that  pro- 
duces rotation  in  the  swinging  follower  arm  is  R±  T$,  perpendicular 
to  I  Ri,  and  this  reduced  to  a  radius  of  I  S±,  equal  to  A  S,  for  purpose 
of  comparison  with  the  cam  rotation,  is  84  T±.     This  value  is  laid 


MISCELLANEOUS   CAM   ACTIONS   AND   CONSTRUCTIONS      217 

off  on  the  4th  ordinate  in  the  velocity  diagram  in  Fig.  173,  as  at 
S^  T±.  In  a  similar  manner  other  values  are  obtained  in  Fig.  173 
and  the  curve  B  Q  C  drawn.  This  curve  shows  the  rate  of  change 
of  angular  velocity  in  the  transmitting  follower  arm  while  the  straight 
horizontal  line  D  E  shews  the  uniform  angular  velocity  of  the  driving 
cam.  The  length  B  C  of  the  base  line  of  the  velocity  diagram  may 
be  taken  any  length  and  then  divided  into  six  equal  parts  to  locate 
the  various  ordinates  of  the  velocity  diagram.  The  length  D  B  in 
Fig.  173  equals  S  T  in  Fig.  172. 

437.  THE  AMOUNT  OF  SLIDING  OF  THE  CAM  may  readily  be  found 
for  example,  by  first  breaking  up  the  velocity  R±  T<z  of  the  point  #4 
on  the  cam  in  Fig.  172  into  its  normal  and  tangential  components — 
the  former  being  shown  at  #4  T5  and  the  latter  at  R±  T& — and,  second, 
by  breaking  up  the  velocity  #4  T$  of  the  corresponding  point  on  the 
swinging  arm  into  the  components  R±  T*>  and  R±  T-j.     The  difference 
TQ  Tr,  in  the  longitudinal  components  will  be  the  rate  of  sliding  at 
that  phase  and  this  difference  is  laid  off  at  S  T  in  Fig.  174.     Sim- 
ilarly other  points  on  the  curve  DTP  are  found.     The  rate  of  sliding 
when  the  circular  surface  of  the  cam  B  F  E  is  in  2  ction,  providing 
there  is  no  stop  rest  for  the  follower  arm,  is  B  D,  equal  to  S  T  in  Fig. 
172;  and  when  the  surface  C  D  is  in  action  it  is  C  F,  Fig.  174. 

438.  A  VELOCITY  CURVE  FOR  THE  FOLLOWER  ROD  N  N' ,  Fig.   172, 

will  give  some  indication  of  its  acceleration  and  retardation  and  the 
relative  strength  of  spring  required  to  operate  it  in  comparison  with 
the  results  secured  by  an  ordinary  roller-end  follower.  The  first 
step  in  this  construction  consists  in  finding  the  six  positions  of  the 
center  M  of  the  upper  curved  surface  Kf  K"  of  the  swinging  arm. 
This  is  readily  done  because  the  points  M,  L  and  /  are  fixed  relatively 
to  each  other,  and,  therefore,  the  point  M±,  for  example,  is  found  by 
taking  /  M  as  a  radius,  /  as  a  center,  and  drawing  an  arc  at  MQ  M$. 
Then  with  L  M  as  a  radius  and  L±  as  a  center  draw  another  short 
arc  intersecting  the  first,  as  at  M±.  With  M  K  as  a  radius  and  M±  as 
a  center  draw  the  arc  passing  through  K±.  The  point  K±  is  on  a  ver- 
tical line  through  M±.  The  horizontal  line  tangent  to  the  arc  at 
K±  will  have  the  position  of  the  bottom  of  the  follower  rod  at  phase  4* 
In  a  similar  way  other  points  on  the  curve  KQ  K&  which  is  the  locus 
of  the  point  of  tangency,  is  obtained.  The  distances  between  the 
horizontal  lines  drawn  through  the  points  KQ,  K\,  etc.,  will  show  the 
amount  of  vertical  travel  of  the  follower  during  each  of  the  six  equal 
time  periods. 


218 


CAMS 


439.  The  velocity  diagram  for  the  vertical  follower  rod  is  quickly 
obtained  by  first  laying  off  the  same  angular  velocity  for  the  swinging 
arm  at  K±,  Fig.  172,  as  was  found  at  R±  and  finding  the  vertical  com- 
ponent of  the  velocity  of  the  point  K±.  This  is  done  in  detail  by  tak- 
ing the  unit  radius  /  $4  together  with  the  linear  velocity  $4  T±  at 
this  unit  radius,  both  of  which  have  already  been  found,  and  trans- 


FIG.  172. — (Duplicate)  SWINGING  TRANSMITTER  ARM  WITH  SLIDING  ACTION 


ferring  the  distance  $4  T±  to  $5  TV  This  will  represent  the  linear 
velocity  of  the  point  85  on  the  radial  line  /  K±.  The  resultant  linear 
velocity  of  the  contact  point  K±  on  the  swinging  arm  is  found,  as 
shown,  to  be  equal  to  K±  Tg.  The  vertical  component  K±  T\Q  of 
this  resultant  velocity  for  the  arm  gives  the  actual  upward  velocity 
of  the  rod  N  N'.  This  value  is  laid  off  at  S  T\Q  in  Fig.  175  and  is  an 
ordinate  on  the  velocity  curve  B  Q  C. 


MISCELLANEOUS    CAM    ACTIONS    AND    CONSTRUCTIONS    219 


440.  The  corresponding  ordinate  S  Us,  Fig. 
175,  for  the  velocity  curve  of  the  follower  rod, 
if  it  had  an  ordinary  roller  end  with  a  roller 
radius  equal  to  B  B'  of  Fig.  172,  may  be  found 
1st,  by  drawing  the  pitch  surface  line  B'  C'  of 
the  cam;  2d,  by  dividing  the  arc  B  P  into  six 
equal  parts;  3d,  by  drawing  a  radial  line  through 
the  fourth  point  P±  to  Bf4 ;  4th,  by  revolving 
B\  to  ^¥4  and  obtaining  the  full  linear  velocity 
N4  ^4  and  laying  it  off  at  B'4  U;  5th,  by  find- 
ing the  radial  velocity  B'±  U%  by  drawing  the 
line  U  C/2  perpendicular  to  the  normal  B\  U\. 
The   length   B \  Ui   will   then    represent   the 
velocity  of  the   follower  bar  if  it  had  a  roller 
end  and  this  length  is  laid  off  at  S  Us  in  Fig. 
175.     Similarly  other  ordinates  of   the  curve 
B  Us  C  are  found. 

441.  COMPARING   THE    VELOCITIES   OF  THE 
FOLLOWER  ROD  N  N',  Fig.  172,  when  a  trans- 
mitting  swinging   arm   is   used   and  when  an 
ordinary  roller  end  is  used,  it  will  be  seen  that 
the  follower  rod  attains  a  higher  velocity  in  the 
former  case  as  shown  by  the  greater  height  of 
the  curve  B  Q  C  over  the  curve  B  Us  C.    Also 
the  acceleration  of  the  follower  rod  N  N'  on  the 
upstroke  will  be  greater  with  the  swinging  arm 
as  is  indicated  by  the  greater  steepness  of  the 
curve  from  B  to  Q  over  that  of  the  curve  from 
B  to  Us. 

442.  THE  SLIDING  ACTION  OF  THE  SURFACE 
K'  K",  Fig.  172,  of  the  swinging  follower  arm 
on  the  bottom  of  the  rod  N  N'  has  a   max- 
imum value  of  about  one- fifth  of  that  of  the 
cam    surface  B  C  on   the  lower  face  of    the 

FIG.  173. — ANGULAR  VELOCITY  DIAGRAM  FOR  CAM  AND 

SWINGING  ARM 

Fio.   174. — SLIDING  VELOCITY  DIAGRAM  OF  CAM  ON  SWING- 
ING ARM  AND  OF  ARM  ON  FOLLOWER  ROD 
FIG.  175. — LINEAR  VELOCITY  OF  FOLLOWER  ROD,  WITH  TRANS- 
MITTING ARM  AND  WITH  ORDINARY  ROLLER  FOLLOWER 
Fio.  176. — PRESSURE  ANGLE  DIAGRAM,  WITH  ORDINARY 
ROLLER  FOLLOWER 


Ed 

£  600-  - 


o 

gsoo- 

0150- 


\ 


B  S±     C 

FIG.  173. 


6001 


FIG.  175. 


FIG.  176. 


220  CAMS 

swinging  arm.  This  is  readily  determined  by  making  use  of  work 
already  done,  as,  for  example,  by  simply  measuring  the  line 
T9  Tio,  in  Fig.  172,  which  is  the  horizontal  or  sliding  component 
of  the  resultant  velocity  K±  TQ  when  the  point  of  driving  contact  is  at 
Kt.  The  distance  Tg  Tio  is  laid  off  at  S  Tn  in  Fig.  174.  Other 
points  of  the  curve  B  Tn  C  are  found  in  the  same  way.  The  ordinates 
of  this  curve  added  to  those  of  the  curve  DTP  would  give  a  measure 
to  the  total  sliding  action  at  any  instant  when  a  swinging  transmitting 
arm  is  used. 

443.  IF  AN  ORDINARY  ROLLER  FOLLOWER  INSTEAD  OF  A  SWINGING 

TRANSMITTING  ARM  WERE  USED  the  pressure  angle  which  would  exist, 
with  a  cam  of  the  size  used  in  Fig.  172  and  with  a  radius  of  roller  equal 
to  B  B',  may  also  be  readily  determined  from  work  already  done. 
For  example,  when  the  center  B\  of  the  roller  is  in  action  the  roller 
will  be  pressing  against  the  cam  in  the  direction  of  the  normal  B '4  Ui 
relatively  to  the  cam  and  the  follower  rod  will  be  moving  in  the  direc- 
tion of  the  radial  line  B\  C/2  relatively  to  the  cam.  Therefore  the 
pressure  angle  at  phase  4  would  be  a,  which  is  equal  to  29°,  and  this 
value  is  laid  off  on  the  fourth  ordinate  as  at  S  V  in  Fig.  176,  thus  ob- 
taining a  point  on  the  pressure  angle  curve  which,  it  will  be  noted, 
has  a  maximum  of  about  31° — a  very  easy  angle  for  general  use. 

444.  IF  IT  IS  DESIRED  TO  KNOW  THE  ACTUAL  RUBBING  VELOCITIES 

IN  FEET  PER  MINUTE  of  the  cam  on  the  swinging  arm,  and  of  the  arm 
on  the  follower  rod ;  and  the  linear  velocity  of  the  follower  rod  N  N', 
Fig.  172,  it  may  quickly  be  obtained  from  the  velocity  diagrams  now 
drawn,  for  any  given  problem.  For  example,  let  it  be  assumed  in 
this  problem  that  the  short  radius  A  B  of  the  cam  in  Fig.  172  is  % 
inch  and  that  the  cam  is  making  900  revolutions  per  minute. 

445.  For  the  data  just  assumed  the  point  B  on  the  cam  will 

.     ..       ,  .75  X  2  X  3.14  X  900      QKQ 

be  moving  with  a  velocity  of  -  — i-^ —  -  =  353  feet  per 

L£ 

minute.  This  then  would  be  the  velocity  represented  by  the  line 
S  T  in  Fig.  172.  Since  all  the  velocity  lines  shown  in  the  drawings 
have  been  found  and  laid  down  without  any  change  in  the  scale  of 
the  drawing,  it  is  only  necessary  to  compute  the  distance  on  S  T  that 
represents  100  feet  per  minute,  and  to  make  that  distance  the  unit 
for  the  velocity  scale  for  measuring  the  curves  in  Figs.  174  and  175. 
If  A  S  measures  %  inch,  S  T  will  be  found  to  measure  .98  inch  to 
the  same  scale.  Then  .98  inch  represents  353  feet  per  minute,  or,  in 
other  words,  .28  inch  represents  100  feet  per  minute.  In  Figs.  174 


MISCELLANEOUS    CAM   ACTIONS   AND    CONSTRUCTIONS    221 

and  175  the  distance  B  A  is  .28  inch  to  the  same  scale  on  which 
A  B  was  measured  in  Fig.  172,  and  this  distance  becomes  the  unit 
measurement  for  100  feet  per  minute  in  the  velocity  diagrams. 

446.  By  drawing  the  scales  as  above  described  it  will  be  noted, 
in  Fig.  174,  that  the  maximum  rubbing  velocity  of  the  cam  on  the 
lower  face  of  the  swinging  arm  is  about  560  feet  per  minute  and  that 
the  maximum  rubbing  velocity  of  the  upper  face  of  the  swinging  arm 
on  the  bottom  of  the  follower  rod  is  about  110  feet  per  minute.     These 
considerations  would  affect  the  design  in  so  far  as  lubrication  and 
wear  are  concerned. 

447.  THE   MAXIMUM   VELOCITY    OF   THE   FOLLOWER   ROD   N  N' 
in  Fig.  172,  may  also  be  read  off  directly  in  Fig.  175,  after  the  scale 
has  been  laid  down  as  above  described.     This  maximum  velocity,  it 
will  be  noted,  is  about  460  feet  per  minute.     Had  an  ordinary  roller 
follower  been  used  on  the  end  of  the  follower  rod,  the  maximum 
velocity  of  the  rod  would  have  been  appreciably  less,  or  about  380 
feet  per  minute.     This  consideration  has  an  important  bearing  on 
strength  of  the  moving  parts  in  the  general  design  of  cam  work.     Its 
comparative  effect,  as  for  example  in  the  strength  of  spring  required 
to  return  the  follower  parts,  may  be  definitely  obtained  by  con- 
structing an  acceleration  and  retardation  diagram  from  the  velocity 
curves  shown  in  Fig.  175,  as  explained  in  detail  in  paragraph  268, 
et  seq. 

448.  BOUNDARY  OF  SURFACE  SUBJECT  TO  WEAR.     In  a  cam  design 
where  there  is  a  sliding  follower  as  in  Fig.  172  it  will  be  of  advantage 
to  know  not  only  the  rubbing  velocities  as  found  above,  but  also  the 
limits  of  the  surfaces  on  which  the  rubbing  takes  place  and  the  posi- 
tions on  the  surfaces  where  the  rubbing  velocities  are  highest  and  the 
pressures  due  to  acceleration  are  greatest.    With  respect  to  the  cam  in 
this  problem,  the  conditions  are  ideal  because  the  accelerations -of 
the  follower  parts  are  greatest  when  the  rubbing  velocities  are  least. 
This  combination  occurs  on  the  portion  of  the  cam  surface  between 
BI  and  #2  as  may  be  pointed  out  as  follows:    (a)  In  Fig.  175  the 
velocity  curve  B  Q  is  steepest  between  the  phases  1  and  2  and  con- 
sequently the  acceleration  of  the  follower  rod  N  N'  is  greatest;   (b), 
In  Fig.  173  where  the  angular  acceleration  of  the  swinging  arm  is 
greatest  also  between  1  and  2\   (c),  and  in  Fig.  174  where  the  sliding 
velocity  is  lowest  between  1  and  2.    The  conditions  for  the  swinging 
arm  I  are  not  so  good.     In  the  first  place  the  total  wear  on  the  lower 
surface  of  the  arm  on  the  upstroke  takes  place  between  J'  and  J3  as 


222  CAMS 

found  by  drawing  the  dashline  arcs  through  the  extremities  R,  #3 
and  RQ  of  the  path  of  action  taking  /  as  a  center  in  each  case;  sec- 
ondly, the  portion  of  the  surface  from  JQ  to  J%  is  rubbed  over  twice  on 
the  upstroke,  or,  in  other  words  it  receives  twice  as  much  wear  as  the 
part  from  J'  to  JQ',  thirdly,  the  rubbing  velocities  are  highest  while 
the  doubly  worn  surface  from  Js  to  JQ  is  in  action  as  indicated  by  the 
higher  part  of  the  curve  from  Q  to  F  in  Fig.  174;  fourthly,  the  part 
of  the  swinging  arm  surface  just  to  the  right  of  JQ  is  also  under  the 
most  intense  pressure,  due  to  acceleration,  as  well  as  being  subjected 
to  double  wear  and  high  velocity,  as  may  be  noted  by  the  fact  that  JQ 
lies  between  the  phases  Ji  and  /2  and  that  between  these  phases  the 
accelerations  are  greatest,  as  indicated  by  the  steepness  of  the  curves 
between  the  ordinates  1  and  2  in  Figs.  173  and  175.  The  points  Ji 
and  /2  are  not  shown  in  Fig.  172,  but  they  may  be  readily  found  by 
drawing  arcs  through  Ri  and  R2  with  I  as  a  center.  The  point  /fe  is 
on  the  path  of  action  just  above  the  point  #2. 

449.  CAM  ACTION  DIFFERENT  ON  UP-AND-DOWN  STROKES.    All  of 
the  velocity  and  sliding  curves  obtained  as  above  for  the  cam  with 
a  transmitting  swinging  arm,  it  will  be  noted,  are  for  the  action  that 
takes  place  while  the  follower  rod  N  Nf,  Fig.  172,  is  on  its  upstroke, 
or,  in  other  words,  while  the  part  of  the  cam  surface  from  B  to  C  is  in 
action.     While  the  follower  is  on  its  downstroke  the  surface  of  the 
cam  from  D  to  E  is  in  action  and  the  velocity  and  the  sliding  curves 
will  be  different,  and  should  be  obtained  by  similar  methods  where 
full  information  for  specific   practical  application    is    desired.     It 
may  easily  happen,  according  to  the   forms  of  the  acting  faces  of 
the  swinging  arm,  that  the  velocities  and  the  accelerations  and  retard- 
ations may  be  quite  different  on  the  two  strokes.     Hence  the  informa- 
tion regarding  both  strokes  should  be  known  in  order  to  properly 
judge  the  friction  and  wearing  characteristics,  and  also  to  judge  the 
strength  of  parts  to  be  used. 

450.  The  disadvantage  of  the  side  pressure  that  accompanies  the 
ordinary  roller-end  follower,  and  the  disadvantage  of  the  high  rubbing 
velocity  that  accompanies  the  swinging  transmitter  arm  which  is 
illustrated  at  I  J  K  in  Fig.  172,  may  be  overcome  by  using  a  roller  on 
the  swinging  arm  to  act  against  the  surface  B  C  of  the  cam,  and  a 
roller  on  the  end  of  the  follower  arm  to  act  on  the  transmitter  head 
at  K'  K".     The  side  pressure  produced  by  the  slope  of  the  cam  is  thus 
taken  up  by  a  tensional  strain  in  the  swinging  arm  instead  of  a  side 
strain  in  the  follower  rod  N  N',  and  a  smoother  and  easier  cam  action 


MISCELLANEOUS    CAM    ACTIONS    AND    CONSTRUCTIONS     223 

should  result  although  there  will  be  an  increased  number  of  parts  in 
the  cam  mechanism. 

451.  PROBLEM  44.     SMALL  CAMS  WITH  SMALL  PRESSURE  ANGLES 
SECURED  BY  USING  VARIABLE  DRIVE.    By  giving  the  cam  shaft  a 
variable  angular  velocity  very  quick  follower  action  may  be  secured 
with  a  relatively  small  cam  without  appreciably  increasing  the  pres- 
sure angle.     To  illustrate,  the  same  data  will  be  taken  as  were  used  in 
Problem  3  except  that  the  follower  is  to  move  up  the  given  3  units  in 
45°  instead  of  90°.     The  complete  statement  of  the  present  problem 
is  as  follows:   Required  a  single  step  radial  cam  to  move  a  follower 
3  units  in  45°  turn  of  the  main  shaft  with  uniform  acceleration  and 
retardation;  to  similarly  return  it  in  the  next  45°,  and  to  allow  it  to 
rest  for  the  remainder  of  the  cycle. 

452.  Let  Nj  Fig.  177,  be  the  center  of  the  uniformly  rotating 
main  shaft  of  the  machine  to  which  the  cam  is  to  be  applied.  Assume 
any  length  for  the  driving  arm  N  P  and  draw  the  two  45°  angles 
P  N  T  and  T  N  Q.     Draw  the  circle  whose  radius  is  N  P  and  divide 
each  of  the  arcs  P  T  and  T  Q  into  six  equal  parts.     Connect  the 
points  Q  and  P,  thus  obtaining  the  point  0  on  N  T  which  will  be 
the  center  of  the  auxiliary  or  cam  shaft.     Attach  a  slotted  arm  0  H 
to  the  cam  shaft,  making  the  shorter  working  radius  of  the  arm  0  J 
equal  to  0  T,  and  the  longer  working  radius  0  H  equal  to  0  N  plus 
N  P.     Assume  the  diameter  of  the  driving  pin  at  P  which  works  in 
the  slotted  arm,  and  make  the  length  of  the  slot  a  little  greater 
than  J  H  to  allow  for  clearance. 

453.  VARIABLE  DRIVE  BY  THE  WHITWORTH  MOTION.     From  the 
preceding  paragraph  it  may  now  be  seen  that  the  arm  0  H,  Fig.  177 
and  the  cam  shaft  to  which  it  is  keyed  will  turn  through  90°  while  the 
main  machine  shaft  turns  through  45°.     The  mechanism  thus  far 
described  for  producing  this  result  is  equivalent  to  the  Whitworth 
slow-advance  and  quick-return  mechanism,  but  any  other  type  of 
slow-advance    and    quick-return    mechanism    that    gives    complete 
rotary  motion  could  be  used  instead. 

454.  To  construct  the  cam,  compute  the  size  of  the  pitch  circle 
in  the  same  manner  as  in  an  elementary  problem,  but  using  the  90° 
that  the  cam  will  turn  during  the  outward  motion  of  the  follower 
instead  of  the  assigned  motion  of  45°  that  the  main  shaft  will  turn. 
Thus  the  diameter  of  the  pitch  circle  will  be  found  to  be, 

3  X  3.46  X  360 


3.14  X  90 


13.2. 


224 


CAMS 


Lay  this  value  off  at  D  S,  Fig.  177,  and  draw  the  pitch  circle  with  0 
as  a  center.  Lay  off  the  assigned  motion  of  3  units  of  the  follower 
symmetrically  about  D  as  at  A  V.  Assuming  6  construction  points 
for  finding  the  cam  pitch  curve,  divide  A  D  into  nine  equal  parts  and 
take  the  1st,  4th  and  9th  division  points;  do  the  same  with  V  D. 
Divide  the  arc  Q  T  into  six  equal  parts  and  draw  radial  lines  through 
each  division  point,  as  indicated  &tOE  and  0  K.  Carry  the  division 


Fia.  177.     PROBLEM  44,  SHOWING  THAT  VERY  SMALL  CAMS  AND  SMALL  PRESSURE  ANGLES 
MAY  BE  OBTAINED  BY  USING  VARIABLE  VELOCITY  DRIVE 


points  on  V  A  around  to  their  corresponding  radial  lines  by  means 
of  circular  arcs,  as  indicated  at  4K\.  Then  the  curve  through  the 
points  A,  KI,  etc.,  will  be  on  the  pitch  surface  of  the  desired  cam. 

455.  The  present  cam  does  the  same  work  in  half  the  time  of 
the  cam  that  is  shown  in  Fig.  32,  and  both  have  the  same  overall 
dimensions.  The  cams  are  of  different  shape,  however.  The  cam 
shaft  0  will  have  widely  varying  angular  velocity,  ranging  between 


MESCELLANEOUS    CAM    ACTIONS    AND    CONSTRUCTIONS     225 


values  which  vary  from  —--  to  — -.     At  the  phase  of  the  median  - 
U  J         U  n. 

ism  shown  by  the  object  lines  in  Fig.  177  the  driving  shaft  N  and 
the  cam  shaft  0  have  the  same  angular  velocity,  and  this  is  true 
for  this  phase  no  matter  what  length  of  driving  arm  is  taken  at  the 
start.  The  cam  will  have  its  greatest  angular  velocity  when  N  P 
is  in  the  position  N  T,  but  at  this  phase  the  pressure  angle  will  be 
zero,  and  it  will  be  comparatively  small  while  the  cam  is  approach- 
ing and  receding  from  this  phase.  Had  a  cam  been  constructed  in 


FIG.   178. — SWASH-PLATE  CAM 

the  regular  way,  that  is  without  variable  drive  of  the  cam  shaft,  to 
give  3  units  motion  in  45°  under  the  condition  of  this  problem  it 
would  have  required  a  cam  with  a  pitch  circle  diameter  of 

3  X  3.46  X  360 
3.14  X  45 

units  instead  of  13.2  as  here  used. 

456.  SWASH-PLATE  CAMS  differ  in  structural  details  from  any 
thus  far  considered  but  they  are,  in  effect,  end  or  cylindrical  cams. 
If  in  Fig.  178  a  basic  cylinder  C  Y  is  intersected  by  an  inclined 
plane  P  L  it  will  cut  a  flat  surface  from  the  cylinder  and  the  form 


226  CAMS 

of  this  surface,  when  viewed  perpendicularly,  will  appear  as  an 
ellipse  in  which  the  major  and  minor  axes  will  be  P  L  and  N  L, 
respectively.  The  flat  surface  thus  formed  is  termed  a  swash  plate. 
It  is  shown  in  the  top  view  by  the  elliptical  curve  PI  L%  and  in  the 
end  view  by  the  circle  PI  LI.  As  the  swash  plate  turns  on  its  axis 
X  Xij  which  is  the  axis  of  the  original  cylinder  it  gives  a  reciprocating 
motion  to  a  follower  rod  F  E.  The  range  of  the  follower  motion  will 
be  greater  or  less  according  to  the  true  radial  distance  AI  D\  of  the 
follower  from  the  axis  of  the  cam,  and  in  the  present  illustration  the 
range  of  follower  motion  is  R  S.  If  a  sharp  F-edge  were  used  on 
the  follower  E  F,  the  contact  would  be  at  A,  instead  of  at  T  as  it  is 
with  the  roller,  and  the  motion  of  the  follower  would  be  harmonic, 
giving  velocity  and  acceleration  curves  similar  to  those  shown  in 
Figs.  88  and  89  respectively.  The  smaller  the  follower  roller  F  T, 
Fig.  178,  the  truer  and  smoother  will  be  the  running  of  the  swash- 
plate  cam. 

457.  ROTARY   CAM   GIVING  INTERMITTENT  ROTARY  MOTION.    A 
cam  of  unusual  form  is  shown  at  A  B  in  Fig.  179.     It  is  designed  to 
change  a  uniform  rotary  motion  in  the  shaft  P  P  to  an  intermittent 
rotary  motion  in  the  shaft  C  by  operating  on  the  roller  pins  D,  E,  F,  G. 
Specifically,  it  is  desired  that  the  shaft  C  shall  make  a  %  turn  while 
the  shaft  P  P  makes  a  }/£  turn,  then  that  the  shaft  C  shall  remain 
stationary  while  P  P  makes  Y%  turn,  and  finally,  that  shaft  C  shall 
be  under  positive  control  all  the  time.     Such  a  cam  would  be  auto- 
matically formed  on  a  previously  prepared  blank  by  using  a  rotary 
cutter  of  the  same  size  as  the  rollers,  D,  E,  etc.,  and  which  travels  in 
the  same  path  as  the  rollers  while  the  cam  blank  is  turned  by  inde- 
pendent means.     For  the  purpose  of  laying  out  the  blank  and  of 
representing  the  form  of  the  cam  surface  in  a  drawing,  the  roller  is 
taken  in  several  intermediate  positions,  one  of  which  is  shown  in  fine 
lines  at  H,  and  constructions  made  as  follows.     The  method  here 
given  is  reduced  to  simplest  terms  and  is  approximate.     It  is  suf- 
ficient, however,  for  the  cam  surface  will  be  true,  because  of  its  auto- 
matic manufacture,  even  if  the  delineation  is  not  exactly  so. 

458.  Make  an  end  view  of  the  roller  as  shown  at  HI,  Fig.  179. 
The  circle  through  HI  represents  the  circle  half  way  down  the  roller 
and  the  short  vertical  line  tangent  to  it  locates  the  point  of  tangency 
for  the  cam  surface  and  roller  assuming  that  the  cam  surface  has  a 
45°  slant  when  the  roller  has  turned  45°  from  E  to  H.     Projecting 
HI  first  to  H  and  then  down  to  HI  on  the  centerline,  a  point  will 


MISCELLANEOUS    CAM    ACTIONS    AND    CONSTRUCTIONS    227 

be  found  on  the  centerline  of  the  cam  surface,  it  being  noted  that 
the  cam  turns  through  90°  while  the  follower  turns  45°.  A  straight 
line  on  the  surface  of  the  roller  through  H  would  represent  approxi- 
mately the  line  of  contact  between  cam  surface  and  roller  and  would 
be  projected  down  to  give  K  and  N  if  the  slant  of  the  cam  surfaces 
edges  were  the  same.  The  slant  for  the  edge  /  K  L  of  larger  radius 
is  a  little  less  than  that  of  M  N  0,  being  on  a  larger  average  radius, 
and,  making  a  corresponding  allowance,  Hz  K  is  taken  a  little  less 


FIG.  179. — SPECIAL  CAM  TYPE  GIVING  INTERMITTENT  ROTARY  MOTION 

than  H2  N.  The  points  L  and  0  will  be  directly  under  the  roller  in 
position  F,  and  the  distances  from  the  centerline  P  P  to  these  points 
will  be  the  same  as  the  distances  from  the  centerline  to  the  corre- 
sponding points  on  the  bottom  of  the  roller.  It  will  be  noted  in  this 
construction  that  the  width  of  the  cam  surface  is  less  than  the  length 
of  the  roller. 

459.  THE  ECCENTRIC  may  be  considered  as  a  special  type  of  cam. 
It  is  widely  used  in  engine  and  other  work  where  it  is  desired  to 
secure  a  simple  reciprocating  motion  from  a  rotary  motion.  Where 


228 


CAMS 


the  initial  motion  may  be  taken  from  the  end  of  a  rotating  shaft,  a 
crank  is  the  simpler  device  to  use,  but  where  the  motion  must  be 
taken  from  an  intermediate  point  on  the  shaft  an  eccentric  is  neces- 


FIG.   180. — PRACTICAL  EXAMPLE  OF  CAM  SHAFT  CARRYING  ELEVEN  CAMS 

sary.  The  eccentric  gives  a  characteristic  motion  to  the  follower 
the  same  as  a  driving  crank  would  give  to  the  crosshead  in  an  ordinary 
crank  and  connecting-rod  mechanism,  the  equivalent  crank  length 
being  equal  to  the  distance  from  the  center  of  the  shaft  to  the  center 


FIG.  181. — SEPARATE  END  VIEWS  OF  CAMS  SHOWN  IN  FIG.  180,  TO  REDUCED  SCALE 

of  the  eccentric  circle  as  shown  at  R  S  in  Fig.  181.  The  eccentric 
cannot  be  used  where  specific  intermediate  velocities  are  desired  for 
the  follower.  The  use  of  the  eccentric  as  a  cam  in  automatic  machin- 
ery is  illustrated  in  Fig.  180  which  represents  the  main  cam  shaft  of  a 
machine  devised  for  special  manufacturing  purposes.  Eleven  cams, 


MISCELLANEOUS    CAM    ACTIONS    AND    CONSTRUCTIONS     229 

compactly  arranged,  are  shown  on  this  shaft,  four  of  them  being 
eccentrics,  namely  Nos.  e,  g,  ti,  and  k.  All  eleven  cams  are  shown  in 
end  view  in  Fig.  181  with  the  exception  of  i,  which  is  shown  to  en- 
larged scale  in  Fig.  179. 

460.  AN  EXAMPLE  OF  A  TIME-CHART  DIAGRAM  for  all  of  the  cams 
illustrated  in  Figs.  180  and  181  is  given  in  Fig.  182.     Time-chart 


FIG.  182. — PRACTICAL  EXAMPLE  OF  TIME  CHART  DIAGRAM  FOB  ELEVEN  CAMS  IN  ONE 

AUTOMATIC  MACHINE 

diagrams  are  treated  in  a  general  way  in  paragraph  19,  and  in  detail 
with  reference  to  a  specific  example  in  paragraphs  143  to  147. 
The  form  of  diagram  here  shown  is  specially  to  be  commended  in 
that  the  individual  diagram  boxes  for  each  cam  are  separated  from 
each  other  by  a  small  space  so  that  it  is  impossible  for  the  heavy 
base  lines  to  touch  or  cross  each  other  under  any  circumstances. 


INDEX 


A  PAGE 

Acceleration  diagrams  for  different 

base  curves 89 

Acceleration     diagrams.     Method 

of  determining 138 

Accelerations  produced  by  differ- 
ent base  curves 142 

Accuracy  in  cam  construction ....   146 

Adjustable  cam  defined 11 

Adjustable  cylindrical  cam  plates. .   193 

All-logarithmic  base  curve 89 

All-logarithmic  cam  problem 94 

Angular  velocity  curve  for  swing- 
ing follower 216 

B 

Balancing  of  cams 148 

Barrel  cam  defined 7 

Base  curve  denned 14 

Base  curves  in  common  use 14 

Base  curves.     Comparison  of.  ...  88 

Base  curves.     Complete  list  of .  .  .  88 
Base    curves.        Construction    of 

common 20 

Base  line  defined 14 

Box  cam  denned. .  8 


Cam  action  different  in  up-and- 
down  strokes 160,  222 

Cam  chart  applied 29 

Cam  chart  defined 12 

Cam  chart  diagram  defined 12 

Cam  considered  as  bent  chart. ...     34 

Cam  defined 1 

Cam    factor    chart    for    all    base 

curves 151 

Cam  factor  chart  for  common  base 

curves 19 

Cam  factors  for  all  base  curves ....   150 
Cam    factors    for    common   base 

curves. .  18 


PAGE 

Cam  factors.     Method  of  deter- 
mining     152 

Cam     mechanism     for     drawing 

ellipse 79 

Cam  mechanism  for  reproducing 

designs 80 

Cam  shaft  acting  as  guide 204 

Cam    size.     Effect    on    pressure 

angle 33 

Cam  surface  on  follower 208,  213 

Cam  with  flat-surface  follower. ...     45 

Cam  with  sliding  follower 57 

Cams  classified 1 

Cams  for  high-speed  work 148 

Cams   for  low-starting  velocities 

129,  132 
Cams  for  swinging  follower  arms 

50,  52,  57 

Carrier  cam  defined 11 

Characteristics  of  base  curves ....     88 

Circles.     Subdivision  of 86 

Circular  base  curve.     Case  1 119 

Circular  cam  problem.     Case  II...   129 

Clamp  cam  defined 11 

Comparison  of  base  curves 88 

Comparison  of  parabola  and  crank 

curves Ill 

Comparison  of  velocities  and  forces 

of  different  base  curves 141 

Conical  cams  denned 2 

Conical  follower  pin  for  cylindrical 

cam 190 

Construction    of    common     base 

curves 20 

Crank  curve  as  projection  of  helix. .  108 
Crank  curve  characteristics. .  .108,  111 

Crank  curve  construction 21 

Cube  base  curve 125 

Cube  curve  cam  problem.     Case  I.  127 
Cube  curve  cam  problem.     Case 

II .133 


231 


232 


INDEX 


PAGE 

Cube  curve  cam  specially  adapted 

for  follower  returned  by  spring  144 

Curved  follower  toe 162 

Cylindrical  cam  defined 1,  7 

Cylindrical  cam  problem 68,  70 

Cylindrical     cams.     Drawing     of 

grooves  in 186 

Cylindrical  cams.     True  pressure 

angle  in 186 

D 

Derived    curve    for   pure    rolling 

action 174 

Diagram.     Cam  chart 12 

Diagram.     Timing 13 

Disk  cam  defined 1 

Dog  cam  defined 11 

Double-acting  cam  defined 9 

Double-disk    positive    drive    cam 

for  swinging  arms 205 

Double-disk  yoke  cam  problem ....  65 

Double-end  cam  defined 7 

Double-mounted  cam  defined.  ...  11 

Double-screw  cams 194 

Double-step  radial  cam 39 

Drum  cam  denned. .  7 


Eccentric  as  a  cam 227 

Ellipse.       Cam     mechanism    for 

drawing  of 79 

Ellipse.     Construction  of 178 

Elliptical    arcs    for    pure    rolling 

action 177 

Elliptical    base    curve    character- 
istics    123 

Elliptical  curve  construction 23 

Empirical  cam  design 25 

End  cam  defined.  .  7 


Face  cam  defined 3 

Face  cam  problem 55 

Factors.     Methods  of  determining 

cam 152 

Factors.     Table  of  cam 18,  150 


PAGE 

Flat-surface  follower 45,  49,  59 

Follower  carrying  cam  surface .  208,  213 
Follower  returned  by  springs  ....  142 
Follower  rollers  for  cylindrical 

cams 188 

Follower  roller.     Size  of 35 

Follower  velocity  in  ft.  per  sec. 

165,  220 

Follower  with  curved  toe 162 

Forces  produced  by  different  base 

curves 141 

Formula  for  cam  size 17 

Frog  cam  defined 2 


Gradual  starting  of  follower  shaft. .  1 77 
Graphical    methods.     Degree    of 

precision  in 141 

Gravity  curve 110 

Groove  cam  defined 7 

H 

Handwriting.       Cam  mechanism 

for  reproducing 79 

Harmonic  curve 108 

Harmonic  motion 108,  206 

Heart  cam  defined 3 

Helix  as  pro  j  ection  of  crank  curv  e . .  1 08 

High  speed  in  cam  work 148 

Hyperbola  for  pure  rolling  action .  184 
Hyperboloidal     follower    pin    for 

cylindrical  cam 190 


Infinite  connecting  rod 108 

Interference  of  cams 75 

Intermediate  transmitter  arm ....  214 

Intermittent  harmonic  motion..  . .  206 

Intermittent  rotary  motion 226 

Internal  cam  defined 8 

Involute  cam  problem 199 

Involute  curve  defined 197 

Involute  curve.     Construction  of.  192 

K 

Keyways.     Location  of 78 


INDEX 


233 


L  PAGE 

Length     of     follower     surface 

58,  62,  159,  164 

Limited  use  of  flat-surface  follow- 
ers  49,  59 

Limited  use  of    single-disk  yoke 

cams 64 

Limiting  size  of  follower  roller 35 

Locus  of  point  of  contact  between 

cam  and  follower 

58,  62,  159,  164,  216 
Logarithmic-combination         cam 

problem 101 

Logarithmic  curve.     Construction 

of 101,  172 

Logarithmic  curve  for  pure  rolling 

action 169,  171 

Logarithmic  curve.     Properties  of  171 
Logarithmic  spiral.     Construction 

of 95,98 

M 

Multiple-mounted  cam  defined ...  11 

Mushroom  cam  defined 3 

Mushroom  cam  problem 45 

N 

Names  of  cams  tabulated 12 

Noise  from  cams 147 

O 

Offset  cam  denned 8 

Offset  cam  problem 42 

Omission  of  cam  chart 31 

Oscillating  cam  defined 11 

Oscillating     single-disk     positive- 
drive  cam. .                             .  202 


Parabola  cam  characteristics 110 

Parabola  construction 22,  182 

Parabola  for  pure  rolling  action.. .  182 

Parabolic  easing-off  arcs 103 

Parabolic  curve.     Property  of . . . .  182 

Perfect  cam  action 110 

Periphery  cam  defined 2 

Pins  for  cylindrical  cams 188 

Pitch  circle  defined .  .  16 


PAGE 

Pitch  line  defined 15 

Pitch  point  defined 16 

Pitch  surface  defined 16 

Plate  cam  defined 3 

Plates  for  cylindrical  cams 193 

Positive-drive  cam  defined 8 

Positive-drive  double-disk  cam  for 

swinging  arms 205 

Positive-drive  single-disk  cam  for 

swinging  arms 202 

Precision  of  graphical  methods .  .  .  141 
Pressure  angle  characteristics  of 

involute  curve 198,  201 

Pressure  angle  defined 16 

Pressure  angle  factors.  .^18,  149,  150 
Pressure  angle  relation  to  cam  size .  31 
Pure  rolling  in  cam  work 168-185 

R 
Radial  cam  defined 1 

Radius  of  curvature  of  non-circular 

arcs 38 

Rate  of  sliding  of  cam  on  surface  of 

follower 164,  166 

Regulation  of  noise  in  cam  design  147 
Relative  strengths  of   springs  re- 
quired for  different  cams ....   143 

Roller.     Limiting  size  of 35 

Rollers  for  cylindrical  cams 188 

Rolling  action 168-185 

Rolling  cam  defined 5 

Rotary  cam  giving  intermittent 

rotary  motion 226 

Rotary  sliding-disk  yoke  cam . .  205,  206 

S 

Scotch  yoke 207 

Screw  cams 193 

Shaft  guide  for  cam  followers ....   204 

Side  cam  defined 1 

Sine  curve 108 

Single-acting  cam  defined 9 

Single-disk  positive  drive  cam  for 

swinging  arms 202 

Single-disk  yoke  cam  problem ....     63 

Single-step  cam  problem 28,  31 

Sinusoid .   108 


234 


INDEX 


PAGE 

Sliding  contact  follower 57 

Sliding  friction  eliminated 168 

Sliding  of  cam  on  follower  surface, 

164,  166,  219 
Slow-advance  and  quick-return  by 

cylindrical  cams 195 

Small  cams  with  small  pressure 
angles    secured    by    variable 

speed  drive 223 

Spherical  cam  defined 2 

Springs.      Use   of,    for   returning 

cam  followers 142 

Starting  velocities  of  cam  followers 

129,  132 

Step  cam  defined 9 

Straight-line  base  curve  construc- 
tion    20 

Straight-line  base  curve  problem .  106 
Straight-line     combination     base 

curve  construction 20,  107 

Straight-sliding  plate  cams 196 

Strap  cam  defined 11 

Subdivision  of  circles 86 

Subtangent  of  logarithmic  curve...  102 

Swash  plate  cam 225 

Swinging  follower  arm 50 

Swinging  transmitter  arm 214 


Table  of  cam  factors  for  all  base 

curves 150 

Tangential  base  curve 113 

Tangential  cam  problem.    Case  I.   113 
Tangential    cam    problem.    Case 

II 135 

Technical  cam  design 27 

Time-acceleration  diagrams 139 

Time  chart  applied 76 

Time  chart  denned . .  13 


PAGE 

Time-chart    diagram    for    eleven 

cams 229 

Time-distance  diagrams 138 

Time-velocity  diagrams 138 

Timing  of  cams.    Problem 75 

Toe-and-wiper  cam  defined 7 

Toe-and-wiper  cam  problem 61 

Toe-and  wiper  cam  with  variable 

angular  velocity 157 

Transmitter  arm  between  cam  and 

follower. .                                .  214 


Variable  angular  velocity  in  cam 

shaft 157 

Variable  speed  for  small  cams ....  223 
Varied  forms  of  fundamental  base 

curves Ill,  149 

Velocities  produced  by  different 

base  curves 141 

Velocity  diagrams  for  different 

base  curves 89 

Velocity  diagrams.  Method  of 

determining 138 

Velocity  of  follower  in  feet  per 

second 165,  220 

W 
Wear.    Distribution  of,  on  follower 

surface 58,  62,  159,  164,  221 

Whitworth  motion 223 

Wiper  cam  defined 5 

Working  surface  defined 16 


Yoke  cam  defined 6 

Yoke  cam  with  rotary  sliding  disk  206 
Yoke  cam  problem.  Double-disk.  65 
Yoke  cam  problem.  Single-disk..  63 


^    ^  I 

Wiley  Special  Subject  Catalogues 

For  convenience  a  list  of  the  Wiley  Special  Subject 
Catalogues,  envelope  size,  has  been  printed.  These 
are  arranged  in  groups — each  catalogue  having  a  key 
symbol.  (See  special  Subject  List  Below).  To 
obtain  any  of  these  catalogues,  send  a  postal  using 
the  key  symbols  of  the  Catalogues  desired. 


I— Agriculture.     Animal  Husbandry.     Dairying.     Industrial 
Canning  and  Preserving. 

2 — Architecture.  :   Building.       Concrete  and  Masonry. 

i 

3 — Business  Administration  and  Management.     Law. 

Industrial  Processes:   Canning  and  Preserving;     Oil  and  Gas 
Production;  Paint;  Printing;  Sugar  Manufacture;  Textile. 

CHEMISTRY 
4a  General;  Analytical,  Qualitative  and  Quantitative;  Inorganic; 

Organic. 
4b  Electro-  and  Physical;  Food  and  Water;  Industrial;  Medical 

and  Pharmaceutical;  Sugar. 
' 

CIVIL  ENGINEERING 

5a  Unclassified  and  Structural  Engineering. 

5b  Materials  and  Mechanics  of  Construction,  including;  Cement 

and     Concrete;    Excavation    and    Earthwork;     Foundations; 

Masonry. 

5c  Railroads;  Surveying. 

5d  Dams;  Hydraulic  Engineering;  Pumping  and  Hydraulics;  Irri- 
gation Engineering;  River  and  Harbor  Engineering;  Water 
Supply. 


CIVIL  ENGINEERING—  Continued 

5e  Highways;  Municipal  Engineering;  Sanitary  Engineering; 
Water  Supply.  Forestry.  Horticulture,  Botany  and 
Landscape  Gardening. 


6 — Design.       Decoration.       Drawing:     General;      Descriptive 
Geometry;  Kinematics;  Mechanical. 

ELECTRICAL  ENGINEERING— PHYSICS 
7 — General  and  Unclassified;  Batteries;  Central  Station  Practice; 
Distribution  and   Transmission;  Dynamo-Electro   Machinery; 
Electro-Chemistry  and   Metallurgy;    Measuring     Instruments 
and  Miscellaneous  Apparatus. 


8 — Astronomy.      Meteorology.      Explosives.      Marine    and 
Naval  Engineering.     Military.     Miscellaneous  Books. 

MATHEMATICS 

9 — General;    Algebra;   Analytic  and   Plane   Geometry;    Calculus; 
Trigonometry;  Vector  Analysis. 

MECHANICAL  ENGINEERING 

lOa  General  and  Unclassified;  Foundry  Practice;  Shop  Practice. 
lOb  Gas  Power  and    Internal   Combustion  Engines;  Heating  and 

Ventilation;  Refrigeration. 
lOc  Machine  Design  and  Mechanism;  Power  Transmission;  Steam 

Power  and  Power  Plants;  Thermodynamics  and  Heat  Power. 
1 1 — Mechanics.  . 

12 — Medicine.  Pharmacy.  Medical  and  Pharmaceutical  Chem- 
istry. Sanitary  Science  and  Engineering.  Bacteriology  and 
Biology. 

MINING  ENGINEERING 

13 — General;  Assaying;  Excavation,  Earthwork,  Tunneling,  Etc.; 
Explosives;  Geology;  Metallurgy;  Mineralogy;  Prospecting; 
Ventilation. 

14 — Food  and  Water.  Sanitation.  Landscape  Gardening. 
Design  and  Decoration.  Housing,  House  Painting. 


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